Model-Based Neuromechanical Controller For A Robotic Leg

ABSTRACT

A model-based neuromechanical controller for a robotic limb having at least one joint includes a finite state machine configured to receive feedback data relating to the state of the robotic limb and to determine the state of the robotic limb, a muscle model processor configured to receive state information from the finite state machine and, using muscle geometry and reflex architecture information and a neuromuscular model, to determine at least one desired joint torque or stiffness command to be sent to the robotic limb, and a joint command processor configured to command the biomimetic torques and stiffnesses determined by the muscle model processor at the robotic limb joint. The feedback data is preferably provided by at least one sensor mounted at each joint of the robotic limb. In a preferred embodiment, the robotic limb is a leg and the finite state machine is synchronized to the leg gait cycle.

RELATED APPLICATIONS

This Application is a divisional of U.S. patent application Ser. No. 12/698,128, filed Feb. 1, 2010, now U.S. Pat. No. 8,864,846, which claims the benefit of U.S. Provisional Patent Appl. Ser. No. 61/148,545, filed Jan. 30, 2009, and is a continuation-in-part of U.S. patent application Ser. No. 12/608,627, filed Oct. 29, 2009, now U.S. Pat. No. 8,870,967, which is a continuation of U.S. patent application Ser. No. 11/642,993, filed Dec. 19, 2006, now abandoned, which claims the benefit of the U.S. Provisional Patent Appl. Ser. No. 60/751,680, filed Dec. 19, 2005, and is a continuation-in-part of U.S. patent application Ser. No. 11/600,291, filed Nov. 15, 2006, now abandoned, which claims the benefit of U.S. Provisional Appl. No. 60/736,929, filed Nov. 15, 2005 and U.S. Provisional Appl. No. 60/705,651, filed Aug. 4, 2005, and is a continuation-in-part of U.S. patent application Ser. No. 11/395,448, filed Mar. 31, 2006, which claims the benefit of U.S. Provisional Patent Appl. No. 60/666,876, filed Mar. 31, 2005, and of U.S. Provisional Patent Appl. No. 60/704,517, filed Aug. 1, 2005, and a continuation-in-part of U.S. patent application Ser. No. 11/499,853, filed Aug. 4, 2006, now U.S. Pat. No. 7,313,463, which claims the benefit of U.S. Provisional Appl. 60/705,651, filed Aug. 4, 2005, now abandoned.

U.S. patent application Ser. No. 11/499,853, filed Aug. 4, 2006, now U.S. Pat. No. 7,313,463 is a continuation-in-part of U.S. patent application Ser. No. 11/395,448, filed Mar. 31, 2006, now abandoned, which claims the benefit of U.S. Provisional Appl. No. 60/704,517, filed Aug. 1, 2005, now expired and U.S. Provisional Appl. No. 60/666,876, filed Mar. 31, 2005, now expired.

U.S. patent application Ser. No. 12/698,128, filed Feb. 1, 2010, now U.S. Pat. No. 8,864,846, is also a continuation-in-part of U.S. patent application Ser. No. 12/157,727, filed Jun. 12, 2008, now U.S. Pat. No. 8,512,415, which claims the benefit of U.S. Provisional Appl. No. 60/934,223, filed Jun. 12, 2007, now expired, and is a continuation-in-part of U.S. patent application Ser. No. 11/642,993, filed Dec. 19, 2006, now abandoned.

The present application claims the benefit of the filing date of each of the foregoing patent applications and incorporates the disclosures of each of the foregoing applications herein by reference in their entirety.

GOVERNMENT SUPPORT

This invention was made with U.S. government support under Grant Numbers VA241-P-0026; 65OD70025 and VA241-P-0479, 650-D85022, awarded by the United States Veterans Administration. The government has certain rights in this invention.

FIELD OF THE TECHNOLOGY

The present invention relates to control of artificial joints and limbs for use in prosthetic, orthotic, exoskeletal, or robotic devices and, in particular, to control methodology for a robotic leg based on a neuromuscular model of locomotion.

BACKGROUND OF THE INVENTION

Legged locomotion of animals and humans is controlled by a complex network of neurons. Proposed in the early 20th century [Brown, T. G., 1914. On the nature of the fundamental activity of the nervous centres; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. J Physiol 48 (1), 18-46.]. and firmly established today [Orlovsky, G., Deliagina, T., Grillner, S., 1999. Neuronal control of locomotion: from mollusc to man. Oxford University Press, New York], the central pattern generator (CPG) forms the basis of this network.

In the current view, the CPG consists of layers of neuron pools in the spinal cord [Rybak, I. A., Shevtsova, N. A., Lafreniere-Roula, M., McCrea, D. A., 2006. Modeling spinal circuitry involved in locomotor pattern generation: insights from deletions during fictive locomotion. J Physiol 577 (Pt 2), 617-639] which, through other neuron pools channeling muscle synergies, provide rhythmic activity to the leg extensor and flexor muscles [Dietz, V., 2003. Spinal cord pattern generators for locomotion. Clin Neurophysiol 114 (8), 1379-1389; Minassian, K., Persy, I., Rattay, F., Pinter, M. M., Kern, H., Dimitrijevic, M. R., 2007. Human lumbar cord circuitries can be activated by extrinsic tonic input to generate locomotor-like activity. Hum Mov Sci 26 (2), 275-295] sufficient to generate stepping movements, even in the absence of spinal reflexes [Grillner, S., Zangger, P., 1979. On the central generation of locomotion in the low spinal cat. Exp Brain Res 34 (2), 241-261; Frigon, A., Rossignol, S., 2006. Experiments and models of sensorimotor interactions during locomotion. Biol Cybern 95 (6), 607-627]. Spinal reflexes are nevertheless part of this complex network [Rybak, I. A., Stecina, K., Shevtsova, N. A., McCrea, D. A., 2006. Modeling spinal circuitry involved in locomotor pattern generation: insights from the effects of afferent stimulation. J Physiol 577 (Pt 2), 641-658], contributing to the selection of locomotive patterns, the timing of the extensor and flexor activities, and the modulation of the CPG output.

Using this combination of a central pattern generation and modulating reflexes, neuromuscular models of lampreys [Ekeberg, O., Grillner, S., 1999. Simulations of neuromuscular control in lamprey swimming. Philos Trans R Soc Lond B Biol Sci 354 (1385), 895-902], salamanders [Ijspeert, A., Crespi, A., Ryczko, D., Cabelguen, J.-M., 2007. From swimming to walking with a salamander robot driven by a spinal cord model. Science 315 (5817), 1416-1420], cats [Ivashko, D. G., Prilutski, B. I., Markin, S. N., Chapin, J. K., Rybak, I. A., 2003. Modeling the spinal cord neural circuitry controlling cat hindlimb movement during locomotion. Neurocomputing 52-54, 621-629; Yakovenko, S., Gritsenko, V., Prochazka, A., 2004. Contribution of stretch reflexes to locomotor control: a modeling study. Biol Cybern 90 (2), 146-155; Maufroy, C., Kimura, H., Takase, K., 2008. Towards a general neural controller for quadrupedal locomotion. Neural Netw 21 (4), 667-681], and humans [Ogihara, N., Yamazaki, N., 2001. Generation of human bipedal locomotion by a bio-mimetic neuro-musculo-skeletal model. Biol Cybern 84 (1), 1-11; Paul, C., Bellotti, M., Jezernik, S., Curt, A., 2005. Development of a human neuro-musculo-skeletal model for investigation of spinal cord injury. Biol Cybern 93 (3), 153-170] have developed into essential tools for studying different control strategies in animal and human locomotion. The emphasis of these models has been to reproduce the architecture of the CPG and underlying reflexes suggested by experiments [Pearson, K., Ekeberg, O., Buschges, A., 2006. Assessing sensory function in locomotor systems using neuro-mechanical simulations. Trends Neurosci 29 (11), 625-631]. However, little attention has been paid to understanding how such architectures might represent or encode principles of locomotion mechanics.

These principles suggest that, in contrast to the complexity of the identified neural networks, legged locomotion requires little or no control. For instance, two conceptual models of walking [Alexander, R., 1976. Mechanics of bipedal locomotion. In: Perspectives in experimental biology (Ed. Davies, P. S.) Pergamon, Oxford; Mochon, S., McMahon, T., 1980. Ballistic walking. J. Biomech. 13 (1), 49-57] and running [Blickhan, R., 1989. The spring-mass model for running and hopping. J. of Biomech. 22, 1217-1227; McMahon, T., Cheng, G., 1990. The mechanism of running: how does stiffness couple with speed? J. of Biomech. 23, 65-78] have been put forth that capture dominant mechanisms of legged locomotion. Researchers have demonstrated the capacity of these models to self-stabilize if the mechanical system is properly tuned [McGeer, T., 1990. Passive dynamic walking. Int. J. Rob. Res. 9 (2), 62-82; McGeer, T., 1992. Principles of walking and running. Vol. 11 of Advances in Comparative and Environmental Physiology. Springer-Verlag Berlin Heidelberg, Ch. 4; Seyfarth, A., Geyer, H., Günther, M., Blickhan, R., 2002. A movement criterion for running. J. of Biomech. 35, 649-655; Ghigliazza, R., Altendorfer, R., Holmes, P., Koditschek, D., 2003. A simply stabilized running model. SIAM J. Applied. Dynamical Systems 2 (2), 187-218]. Walking and running robots have moreover demonstrated the practical relevance and control benefits derived from this principle [Raibert, M., 1986. Legged robots that balance. MIT press, Cambridge; McGeer, T., 1990. Passive dynamic walking. Int. J. Rob. Res. 9 (2), 62-82; Saranli, U., Buehler, M., Koditschek, D., 2001. Rhex: A simple and highly mobile hexapod robot. Int. Jour. Rob. Res. 20 (7), 616-631; Collins, S., Ruina, A., Tedrake, R., Wisse, M., 2005. Efficient bipedal robots based on passive-dynamic walkers. Science 307 (5712), 1082-1085]. But it remains an open question how this and other principles of legged mechanics are integrated into the human motor control system.

The importance of this interplay between mechanics and motor control has been recognized by neuroscientists and biomechanists alike [Pearson, K., Ekeberg, O., Buschges, A., 2006. Assessing sensory function in locomotor systems using neuro-mechanical simulations. Trends Neurosci 29 (11), 625-631]. For instance, although it is generally accepted that the CPG forms a central drive for motor activity in locomotion [Grillner, S., Zangger, P., 1979. On the central generation of locomotion in the low spinal cat. Exp Brain Res 34 (2), 241-261; Dietz, V., 2003. Spinal cord pattern generators for locomotion. Clin Neurophysiol 114 (8), 1379-1389; Frigon, A., Rossignol, S., 2006. Experiments and models of sensorimotor interactions during locomotion. Biol Cybern 95 (6), 607-627; Ijspeert, A. J., 2008. Central pattern generators for locomotion control in animals and robots: a review. Neural Netw 21 (4), 642-653], Lundberg suggested in 1969 that, out of its rather simple central input, spinal reflexes, which relay information about locomotion mechanics, could shape the complex muscle activities seen in real locomotion [Lundberg, A., 1969. Reflex control of stepping. In: The Nansen memorial lecture V, Oslo: Universitetsforlaget, 5-42]. Refining this idea, Taga later proposed that, because “centrally generated rhythms are entrained by sensory signals which are induced by rhythmic movements of the motor apparatus . . . [,] motor output is an emergent property of the dynamic interaction between the neural system, the musculo-skeletal system, and the environment” [Taga, G., 1995. A model of the neuro-musculo-skeletal system for human locomotion. I. Emergence of basic gait. Biol. Cybern. 73 (2), 97-111]. In support, he presented a neuromuscular model of human locomotion that combines a CPG with sensory feedback and demonstrates how basic gait can emerge from the global entrainment between the rhythmic activities of the neural and of the musculo-skeletal system.

What the actual ratio of central and reflex inputs is that generates the motor output continues to be debated [Pearson, K. G., 2004. Generating the walking gait: role of sensory feedback. Prog Brain Res 143, 123-129; Frigon, A., Rossignol, S., 2006. Experiments and models of sensorimotor interactions during locomotion. Biol Cybern 95 (6), 607-627; Hultborn, H., 2006. Spinal reflexes, mechanisms and concepts: from Eccles to Lundberg and beyond. Prog Neurobiol 78 (3-5), 215-232; Prochazka, A., Yakovenko, S., 2007. The neuromechanical tuning hypothesis. Prog Brain Res 165, 255-265]. For instance, for walking cats, it has been estimated that only about 30 percent of the muscle activity observed in the weight bearing leg extensors can be attributed to muscle reflexes [Prochazka, A., Gritsenko, V., Yakovenko, S., 2002. Sensory control of locomotion: reflexes versus higher-level control. Adv Exp Med Biol 508, 357-367; Donelan, J. M., McVea, D. A., Pearson, K. G., 2009. Force regulation of ankle extensor muscle activity in freely walking cats. J Neurophysiol 101 (1), 360-371].

In humans, the contribution of reflexes to the muscle activities in locomotion seems to be more prominent. Sinkjaer and colleagues estimated from unloading experiments that reflexes contribute about 50 percent to the soleus muscle activity during stance in walking [Sinkjaer, T., Andersen, J. B., Ladouceur, M., Christensen, L. O., Nielsen, J. B., 2000. Major role for sensory feedback in soleus EMG activity in the stance phase of walking in man. J Physiol 523 Pt 3, 817-827]. More recently, Grey and colleagues found that the soleus activity changes proportionally to changes in the Achilles tendon force, suggesting a direct relationship between positive force feedback and activity for this muscle [Grey, M. J., Nielsen, J. B., Mazzaro, N., Sinkjaer, T., 2007. Positive force feedback in human walking. J Physiol 581 (1), 99-105]. Whether such a large reflex contribution is present for all leg muscles remains open. Perhaps a proximo-distal gradient exists in motor control where proximal leg muscles are mainly controlled by central inputs while distal leg muscles are dominated by reflex inputs due to higher proprioceptive feedback gains and a larger sensitivity to mechanical effects, as Daley and colleagues concluded from locomotion experiments with birds [Daley, M. A., Felix, G., Biewener, A. A., 2007. Running stability is enhanced by a proximo-distal gradient in joint neuromechanical control. J Exp Biol 210 (Pt 3), 383-394].

Adaptation to terrain is an important aspect of walking. Today's commercially-available ankle-foot prostheses utilize lightweight, passive structures that are designed to present appropriate elasticity during the stance phase of walking [S. Ron, Prosthetics and Orthotics: Lower Limb and Spinal. Lippincott Williams & Wilkins 2002]. The advanced composites used in these devices permit some energy storage during controlled dorsiflexion and plantar flexion, and subsequent energy release during powered plantar flexion, much like the Achilles tendon in the intact human [A. L. Hof, B. A. Geelen, Jw. Van den Berg, “Calf muscle moment, work and efficiency in level walking; role of series elasticity,” Journal of Biomechanics, Vol. 16, No. 7, pp. 523-537, 1983; D. A. Winter, “Biomechanical motor pattern in normal walking,” Journal of Motor Behavior, Vol. 15, No. 4, pp. 302-330, 1983].

Although this passive-elastic behavior is a good approximation to the ankle's function during slow walking, normal and fast walking speeds require the addition of external energy, and thus cannot be implemented by any passive ankle-foot device [M. Palmer, “Sagittal plane characterization of normal human ankle function across a range of walking gait speeds,” Master's Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 2002; D. H. Gates, “Characterizing ankle function during stair ascent, descent, and level walking for ankle prosthesis and orthosis design,” Master's Thesis, Boston University, 2004; A. H. Hansen, D. S. Childress, S. C. Miff, S. A. Gard, K. P. Mesplay, “The human ankle during walking: implication for the design of biomimetic ankle prosthesis,” Journal of Biomechanics, Vol. 37, Issue 10, pp. 1467-1474, 2004]. This deficiency is reflected in the gait of transtibial amputees using passive ankle-foot prostheses. Their self-selected walking speed is slower, and stride length shorter, than normal [D. A. Winter and S. E. Sienko. “Biomechanics of below-knee amputee gait,” Journal of Biomechanics, 21, pp. 361-367, 1988]. In addition, their gait is distinctly asymmetric: the range of ankle movement on the unaffected side is smaller [H. B. Skinner and D. J. Effeney, “Gait analysis in amputees,” Am J Phys Med, Vol. 64, pp. 82-89, 1985; H. Bateni and S. Olney, “Kinematic and kinetic variations of below-knee amputee gait,” Journal of Prosthetics & Orthotics, Vol. 14, No. 1, pp. 2-13, 2002], while, on the affected side, the hip extension moment is greater and the knee flexion moment is smaller [D. A. Winter and S. E. Sienko.“Biomechanics of below-knee amputee gait,” Journal of Biomechanics, 21, pp. 361-367, 1988; H. Bateni and S. Olney, “Kinematic and kinetic variations of below-knee amputee gait,” Journal of Prosthetics & Orthotics, Vol. 14, No. 1, pp. 2-13, 2002]. They also expend greater metabolic energy walking than non-amputees [N. H. Molen, “Energy/speed relation of below-knee amputees walking on motor-driven treadmill,” Int. Z. Angew, Physio, Vol. 31, p 173, 1973; G. R. Colborne, S. Naumann, P. E. Longmuir, and D. Berbrayer, “Analysis of mechanical and metabolic factors in the gait of congenital below knee amputees,” Am. J. Phys. Med. Rehabil., Vol. 92, pp 272-278, 1992; R. L. Waters, J. Perry, D. Antonelli, H. Hislop. “Energy cost of walking amputees: the influence of level of amputation,” J Bone Joint Surg. Am., Vol. 58, No. 1, pp. 4246, 1976; E. G. Gonzalez, P. J. Corcoran, and L. R. Rodolfo. Energy expenditure in B/K amputees: correlation with stump length. Archs. Phys. Med. Rehabil. 55, 111-119, 1974; D. J. Sanderson and P. E. Martin. “Lower extremity kinematic and kinetic adaptations in unilateral below-knee amputees during walking,” Gait and Posture. 6, 126 136, 1997; A. Esquenazi, and R. DiGiacomo. “Rehabilitation After Amputation,” Journ Am Podiatr Med Assoc, 91(1): 13-22, 2001]. These differences could possibly be a result of the amputees' greater use of hip power to compensate for the lack of ankle power [A. D. Kuo, “Energetics of actively powered locomotion using the simplest walking model,” J Biomech Eng., Vol. 124, pp. 113-120, 2002; A. D. Kuo, J. M. Donelan, and A. Ruina, “Energetic consequences of walking like an inverted pendulum: Step-sto-step transitions,” Exerc. Sport Sci. Rev., Vol. 33, No. 2, pp. 88-97, 2005; A. Ruina, J. E. Bertram, and M. Srinivasan, “A collisional model of the energetic cost of support work qualitatively explains leg sequencing in walking and galloping, pseudo-elastic leg behavior in running and the walk-to-run transition.” J. Theor. Biol., Vol. 237, No. 2, pp. 170-192, 2005].

Passive ankle-foot prostheses cannot provide the capability of adaptation to terrain. To provide for a normal, economical gait beyond slow walking speeds, powered ankle-foot prostheses have now been developed [S. Au and H. Herr. “Initial experimental study on dynamic interaction between an amputee and a powered ankle-foot prosthesis,” Workshop on Dynamic Walking: Mechanics and Control of Human and Robot Locomotion, Ann Arbor, Mich., May 2006; S. K. Au, J. Weber, and H. Herr, “Biomechanical design of a powered ankle-foot prosthesis,” Proc. IEEE Int. Conf. On Rehabilitation Robotics, Noordwijk, The Netherlands, pp. 298-303, June 2007; S. Au, J. Weber, E. Martinez-Villapando, and H. Herr. “Powered Ankle-Foot Prosthesis for the Improvement of Amputee Ambulation,” IEEE Engineering in Medicine and Biology International Conference. August 23-26, Lyon, France, pp. 3020-3026, 2007; H. Herr, J. Weber, S. Au. “Powered Ankle-Foot Prosthesis,” Biomechanics of the Lower Limb in Health, Disease and Rehabilitation. September 3-5, Manchester, England, pp. 72-74, 2007; S. K. Au, “Powered Ankle-Foot Prosthesis for the Improvement of Amputee Walking Economy,” Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 2007; S. Au, J. Weber, and H. Herr. “Powered Ankle-foot Prosthesis Improves Walking Metabolic Economy,” IEEE Trans. on Robotics, Vol. 25, pp. 51-66, 2009; J. Hitt, R. Bellman, M. Holgate, T. Sugar, and K. Hollander, “The sparky (spring ankle with regenerative kinetics) projects: Design and analysis of a robotic transtibial prosthesis with regenerative kinetics,” in Proc. IEEE Int. Conf. Robot. Autom., Orlando, Fla., pp 2939-2945, May 2006; S. K. Au, H. Herr, “On the Design of a Powered Ankle-Foot Prosthesis: The Importance of Series and Parallel Elasticity,” IEEE Robotics & Automation Magazine. pp. 52-59, September 2008]. Some of these are of size and weight comparable to the intact human ankle-foot, and have the elastic energy storage, motor power, and battery energy to provide for a day's typical walking activity [S. K. Au, H. Herr, “On the Design of a Powered Ankle-Foot Prosthesis: The Importance of Series and Parallel Elasticity,” IEEE Robotics & Automation Magazine. pp. 52-59, September 2008].

The use of active motor power in these prostheses raises the issue of control. In previous work with these powered devices, the approach taken was to match the torque-ankle state profile of the intact human ankle for the activity to be performed [S. K. Au, “Powered Ankle-Foot Prosthesis for the Improvement of Amputee Walking Economy,” Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 2007; J. Hitt, R. Bellman, M. Holgate, T. Sugar, and K. Hollander, “The sparky (spring ankle with regenerative kinetics) projects: Design and analysis of a robotic transtibial prosthesis with regenerative kinetics,” in Proc. IEEE Int. Conf. Robot. Autom., Orlando, Fla., pp 2939-2945, May 2006; F. Sup, A. Bohara, and M. Goldfarb, “Design and Control of a Powered Transfemoral Prosthesis,” The International Journal of Robotics Research, Vol. 27, No. 2, pp. 263-273, 2008]. The provision of motor power meant that the open work loops of the angle-torque profiles in faster walking could be supported, rather than just the spring-like behavior provided by passive devices. However, this control approach exhibited no inherent adaptation. Instead, torque profiles were required for all intended activities and variation of terrain, along with an appropriate means to select among them.

In general, existing commercially available active ankle prostheses are only able to reconfigure the ankle joint angle during the swing phase, requiring several strides to converge to a terrain-appropriate ankle position at first ground contact. Further, they do not provide any of the stance phase power necessary for normal gait, and therefore cannot adapt net stance work with terrain slope. In particular, control schemes for powered ankle-foot prostheses rely upon fixed torque-ankle state relationships obtained from measurements of intact humans walking at target speeds and across known terrains. Although effective at their intended gait speed and terrain, these controllers do not allow for adaptation to environmental disturbances such as speed transients and terrain variation.

Neuromuscular models with a positive force feedback reflex scheme as the basis of control have recently been employed in simulation studies of the biomechanics of legged locomotion [H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication); H. Geyer, A. Seyfarth, R. Blickhan, “Positive force feedback in bouncing gaits?,” Proc. R Society. Lond. B 270, pp. 2173-2183, 2003]. Such studies show promise regarding the need for terrain adaptation.

SUMMARY OF THE INVENTION

In one aspect, the present invention is a controller and a control methodology for a biomimetic robotic leg based on a neuromuscular model of human locomotion. The control architecture commands biomimetic torques at the ankle, knee, and hip joints of a powered leg prosthesis, orthosis, or exoskeleton during walking In a preferred embodiment, the powered device includes artificial ankle and knee joints that are torque controllable. Appropriate joint torques are provided to the user as determined by the feedback information provided by sensors mounted at each joint of the robotic leg device. These sensors include, but are not limited to, angular joint displacement and velocity using digital encoders, hall-effect sensors or the like, torque sensors at the ankle and knee joints, and at least one inertial measurement unit (IMU) located between the knee and the ankle joints.

Sensory information of joint state (position and velocity) from the robotic leg is used as inputs to a neuromuscular model of human locomotion. Joint state sensory information from the robotic leg is used to determine the internal state for each of its virtual muscles, and what the individual virtual muscle force and stiffness should be given particular levels of muscle activation is determined from a spinal reflex model. If the robotic leg is a leg prosthesis worn by a transfemoral amputee, angular sensors at the ankle and knee measure joint state for these joints. For the hip joint, the absolute orientation of the user's thigh is determined using both the angular joint sensor at the prosthetic knee and an IMU positioned between the prosthetic knee and the ankle joints. To estimate hip position and velocity, the control architecture works under the assumption that the upper body (torso) maintains a relative vertical position during gait.

In one aspect, the invention is a model-based neuromechanical controller for a robotic limb comprising at least one joint, the controller comprising a neuromuscular model including a muscle model, muscle tendon lever arm and muscle tendon length equations and reflect control equations, the neuromuscular model being configured to receive feedback data relating to a measured state of the robotic limb and, using the feedback data, and the muscle model, muscle tendon lever arm and muscle tendon length equations and reflect control equations of the neuromuscular model, to determine at least one torque command, the controller further comprising a torque control system in communication with the neuromuscular model, whereby the torque control system receives the at least one torque command from the neuromuscular model for controlling the robotic limb joint. In a preferred embodiment, the feedback data is provided by at least one sensor mounted at each joint of the robotic limb. In another preferred embodiment, the robotic limb is a leg and the controller further includes a finite state machine synchronized to the leg gait cycle, the finite state machine being configured to receive the feedback data from the at least one sensor to determine a gait phase of the robotic leg using the feedback data received.

In another aspect, the invention is a model-based method for controlling a robotic limb comprising at least one joint, comprising the steps of receiving feedback data relating to the state of the robotic limb at a finite state machine, determining the state of the robotic limb using the finite state machine and the received feedback data, determining, using a neuromuscular model that includes muscle tendon lever arm and muscle tendon length equations and reflex control equations, and state information from the finite state machine, at least one desired joint torque or stiffness command to be sent to the robotic limb and commanding the biomimetic torques and stiffnesses determined by the muscle model processor at the robotic limb joint.

BRIEF DESCRIPTION OF THE DRAWINGS

Other aspects, advantages and novel features of the invention will become more apparent from the following detailed description of the invention when considered in conjunction with the accompanying drawings wherein:

FIG. 1 is a block diagram of an exemplary embodiment of a general neuromuscular model architecture, according to one aspect of the present invention;

FIGS. 2A-F depict six stages in the evolution of a general neuromuscular model architecture, according to one aspect of the present invention;

FIG. 3 graphically depicts pattern generation, according to one aspect of a general neuromuscular model architecture according to the present invention;

FIGS. 4A and 4B depict walking of a human model self-organized from dynamic interplay between model and ground, and the corresponding ground reaction force, respectively, according to one aspect of the present invention;

FIGS. 5A-C compare steady state walking for the model and a human subject for hip, knee, and ankle, respectively, according to one aspect of the present invention;

FIGS. 6A-D depict adaptation to walking up stairs, including snapshots of the model (FIG. 6A), net work (FIG. 6B), extensor muscle activation patterns (FIG. 6C), and the corresponding ground reaction force (FIG. 6D), according to one aspect of the present invention;

FIGS. 7A-D depict adaptation to walking down stairs, including snapshots of the model (FIG. 7A), net work (FIG. 7B), extensor muscle activation patterns (FIG. 7C), and the corresponding ground reaction force (FIG. 7D), according to one aspect of the present invention;

FIG. 8 is a schematic of a muscle-tendon model, according to one aspect of the present invention;

FIGS. 9A-C depict a contact model, according to one aspect of the present invention;

FIGS. 10A-C depict an exemplary embodiment of an ankle-foot prosthesis used in a preferred embodiment, depicting the physical system (FIG. 10A), a diagram of the drive train (FIG. 10B), and a mechanical model (FIG. 10C), respectively, according to one aspect of the present invention;

FIG. 11 is a diagram of an exemplary embodiment of a finite state machine synchronized to the gait cycle, with state transition thresholds and equivalent ankle-foot biomechanics during each state, used to implement top level control of the ankle-foot prosthesis of FIGS. 10A-C, according to one aspect of the present invention;

FIG. 12 is a block diagram of an exemplary embodiment of a control system for an ankle-foot prosthesis, according to one aspect of the present invention;

FIGS. 13A-C are exemplary plots of prosthesis torque over one complete gait cycle for three walking conditions: level-ground (FIG. 13A), ramp ascent (FIG. 13B), and ramp descent (FIG. 13C), according to one aspect of the present invention;

FIGS. 14A-C depict an exemplary embodiment of the musculoskeletal model as implemented on the prosthetic microcontroller, including the two-link ankle joint model (FIG. 14A), detailed Hill-type muscle model (FIG. 14B), and geometry of the muscle model skeletal attachment (FIG. 14C), according to one aspect of the present invention;

FIG. 15 depicts an exemplary embodiment of a reflex scheme for the virtual plantar flexor muscle, including the relationship among ankle angle, muscle force, and the plantar flexor component of ankle torque, according to one aspect of the present invention;

FIGS. 16A and 16B depict prosthesis-measured torque and angle trajectories during trials with an amputee subject compared to those of the biological ankle of a weight and height-matched subject with intact limbs, including ankle torque and ankle angle, respectively;

FIG. 17 is a comparison of the torque profile after parameter optimization to the biologic torque profile, according to one aspect of the present invention; and

FIGS. 18A-C are plots of experimentally measured prosthesis torque-angle trajectories for an exemplary embodiment of the invention for three different walking conditions: level ground (FIG. 18A), ramp ascent (FIG. 18B), and ramp descent (FIG. 18C).

DETAILED DESCRIPTION

A control architecture is presented to command biomimetic torques at the ankle, knee, and hip joints of a powered leg prosthesis, orthosis, or exoskeleton during walking. In this embodiment, the powered device includes artificial ankle and knee joints that are torque controllable. Appropriate joint torques are provided to the user as determined by the feedback information provided by sensors mounted at each joint of the robotic leg device. These sensors include, but are not limited to, angular joint displacement and velocity using digital encoders, hall-effect sensors or the like, torque sensors at the ankle and knee joints and at least one inertial measurement unit (IMU) located between the knee and the ankle joints.

Sensory information of joint state (position and velocity) from the robotic leg (hip, knee and ankle) is used as inputs to a neuromuscular model of human locomotion. This model uses joint state sensory information from the robotic leg to determine the internal state for each of its virtual muscles, and establishes what the individual virtual muscle force and stiffness should be given particular levels of muscle activation determined from a spinal reflex model. If the robotic leg is a leg prosthesis worn by a transfemoral amputee, angular sensors at the ankle and knee measure joint state for these joints. For the hip joint, the absolute orientation of the user's thigh is determined using both the angular joint sensor at the prosthetic knee and an IMU positioned between the prosthetic knee and the ankle joints. To estimate hip position and velocity, the control architecture works under the assumption that the upper body (torso) maintains a relative vertical position during gait.

As used herein, and in the Appl.s incorporated by reference herein, the following terms expressly include, but are not to be limited to:

“Actuator” means a type of motor, as defined below.

“Agonist” means a contracting element that is resisted or counteracted by another element, the antagonist.

“Agonist-antagonist actuator” means a mechanism comprising (at least) two actuators that operate in opposition to one another: an agonist actuator that, when energized, draws two elements together and an antagonist actuator that, when energized, urges the two elements apart.

“Antagonist” means an expanding element that is resisted or counteracted by another element, the agonist.

“Biomimetic” means a man-made structure or mechanism that mimics the properties and behavior of biological structures or mechanisms, such as joints or limbs.

“Dorsiflexion” means bending the ankle joint so that the end of the foot moves upward.

“Elastic” means capable of resuming an original shape after deformation by stretching or compression.

“Extension” means a bending movement around a joint in a limb that increases the angle between the bones of the limb at the joint.

“Flexion” means a bending movement around a joint in a limb that decreases the angle between the bones of the limb at the joint.

“Motor” means an active element that produces or imparts motion by converting supplied energy into mechanical energy, including electric, pneumatic, or hydraulic motors and actuators.

“Plantarflexion” means bending the ankle joint so that the end of the foot moves downward.

“Spring” means an elastic device, such as a metal coil or leaf structure, which regains its original shape after being compressed or extended.

An exemplary embodiment of a neuromuscular model-based control scheme according to this aspect of the invention is shown as a block diagram in FIG. 1. In FIG. 1, a neuromuscular model 100 according to the invention includes reflex control equations 110 for each modeled muscle unit 120. The predicted forces and stiffnesses from all the modeled muscles are used to compute 130 model estimates of desired joint torques and stiffnesses using muscle moment arm values provided by muscle tendon lever arm and muscle tendon length equations 140, which moment arm and muscle tendon length values from joint state, e.g., joint angle, and data from the literature. A muscle tendon unit (MTU), also referred to a muscle tendon complex (MTC), and associated parameters, are described below with reference to FIGS. 8, 14 and 15. For the purposes of this description, MTU has the same meaning as MTC. The model estimates are then sent to torque control system of biomimetic robotic leg 150 as desired net torque and stiffness values for joints of biomimetic robotic leg 150. Top level finite state machine 160 then tracks the torque and stiffness values at each robotic joint of biomimetic robotic leg 150. Finite state machine 160 receives joint state as an input and provides gait phase and joint state as outputs to neuromuscular model 100.

In order for each of the virtual muscle to produce its required force, a muscle stimulation parameter STIM(t) is required. This parameter can be determined from either an outside input or a local feedback loop. In the control methodology for the exemplary biomimetic leg, the STIM(t) is computed based on local feedback loops. This architecture is based on the reflex feedback framework developed by Geyer and Herr [H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication), herein incorporated by reference in its entirety]. In this framework the neural-control is designed to mimic the stretch reflex of an intact human muscle. This neuromuscular reflex-based control methodology allows the biomimetic robotic leg to replicate human-like joint mechanics.

Neuromechanical model. A human model with a reflex control that encodes principles of legged mechanics predicts human walking dynamics and muscle activities. While neuroscientists identify increasingly complex neural networks that control animal and human gait, biomechanists find that locomotion requires little motor control if principles of legged mechanics are heeded. Here it is shown how muscle reflex behavior could be vital to link these two observations. A model of human locomotion was developed that is driven by muscle reflex behaviors that encode principles of legged mechanics. Equipped with this principle-based reflex control, the model stabilizes into the walking gait from its dynamic interplay with the ground, tolerates ground disturbances, and self-adapts to stairs. Moreover, the model shows qualitative agreement with joint angles, joint torques and muscle activations known from experiments, suggesting that human motor output could largely be shaped by muscle reflex behaviors that link principles of legged mechanics into the neural networks responsible for locomotion.

A human walking model with a motor control is based on muscle reflexes, which are designed to include such principles of legged mechanics. These principles derive from simple conceptual models of legged locomotion and include the reliance on compliant leg behavior in stance [Blickhan, R., 1989. The spring-mass model for running and hopping. J. of Biomech. 22, 1217-1227; Ghigliazza, R., Altendorfer, R., Holmes, P., Koditschek, D., 2003. A simply stabilized running model. SIAM J. Applied. Dynamical Systems 2 (2), 187-218; Geyer, H., Seyfarth, A., Blickhan, R., 2006. Compliant leg behaviour explains the basic dynamics of walking and running. Proc. R. Soc. Lond. B 273, 2861-2867], the stabilization of segmented legs based on static joint torque equilibria [Seyfarth, A., Günther, M., Blickhan, R., 2001. Stable operation of an elastic three-segmented leg. Biol. Cybern. 84, 365-382; Günther, M., Keppler, V., Seyfarth, A., Blickhan, R., 2004. Human leg design: optimal axial alignment under constraints. J. Math. Biol. 48, 623-646], the exploitation of ballistic swing-leg mechanics [Mochon, S., McMahon, T., 1980. Ballistic walking. J. Biomech. 13 (1), 49-57], and the enhancement of gait stability using swing-leg retraction [Seyfarth, A., Geyer, H., Günther, M., Blickhan, R., 2002. A movement criterion for running. J. of Biomech. 35, 649-655; Seyfarth, A., Geyer, H., Herr, H. M., 2003. Swing-leg retraction: a simple control model for stable running. J Exp. Biol. 206, 2547-2555]. Hill-type muscles combined with spinal reflexes are employed, including positive force and length feedback schemes, to effectively encode these mechanical features.

Comparing the model's behavior with kinetic, kinematic, and electromyographic evidence from the literature for human walking, it has been shown that a neuromuscular model with a motor control designed to encode principles of legged mechanics can produce biological walking mechanics and muscle activities. This reflex control allows the model to tolerate sudden changes in ground level and to adapt to stair ascent and descent without parameter interventions.

The structure and control of the human model evolves in six steps from a conceptual point-mass model into a neuromuscular biped with an upper body and two, three-segment legs each actuated by seven muscles and controlled by muscle reflexes. FIGS. 2A-F depict six stages in the evolution of a general neuromuscular model architecture, according to this aspect of the present invention. The first three stages integrate and stabilize compliant leg behavior in stance (FIG. 2A-C). The fourth stage adds an upper body and its balance control (FIG. 2D). The last two stages prepare and ensure the pro- and retraction of the legs during swing (FIGS. 2E and 2F).

In FIGS. 2A-F, described in more detail in the paragraphs that follow, evolving from a stance leg configuration (FIG. 2A), compliant leg behavior as key to walk and run is generated (FIG. 2B) by driving the soleus muscle (SOL) and the lumped vasti group muscles (VAS) with positive force feedbacks F+. To prevent knee overextension the biarticular gastrocnemius muscle (GAS) is added (FIG. 2C) using F+, and the VAS gets inhibited if the knee extends beyond a 170° threshold. To prevent ankle overextension, the tibialis anterior muscle (TA) is added whose pulling of the ankle joint into a flexed position by positive length feedback L+ is suppressed under normal stance conditions by negative force feedback F− from soleus. To allow leg swings, an upper body is added (FIG. 2D). It is driven into a reference lean with respect to the vertical by the hip flexor (HFL) and co-activated hip extensor muscles (GLU, HAM) of the stance leg, where the biarticular HAM prevents knee overextension resulting from hip extensor moments. The landing of the other (leading) leg initiates swing by adding/subtracting a constant stimulation to HFL/GLU, respectively, and by suppressing VAS proportionally to the load borne by the other leg (FIG. 2E). The actual leg swing is facilitated by HFL using L+ until it gets suppressed by L− of HAM (FIG. 2F). HFL's stimulation is biased dependent on the upper body's lean at take-off. Moreover, using F+ for GLU and HAM retracts and straightens the leg toward the end of swing. Finally, the now unsuppressed L+ of TA drives the ankle to a flexed position (FIG. 2G).

Stance leg compliance and stability. The bipedal spring-mass model is used as the starting point for the conceptual basis for human locomotion (FIG. 2A). Although this model consists only of point-mass 205 that progresses on two massless spring legs 210, 215, it reproduces the center of mass dynamics observed in human walking and running, unifying both gaits in one conceptual framework based on compliant leg behavior in stance [Geyer, H., Seyfarth, A., Blickhan, R., 2006. Compliant leg behaviour explains the basic dynamics of walking and running. Proc. R. Soc. Lond. B 273, 2861-2867].

To implement compliant behavior in neuromuscular legs, each spring 210, 215 is replaced with thigh 220, shank 225, and foot 230, and a soleus muscle (SOL) 235 and a vasti muscle group (VAS) 240 are added, both generating their muscle activity through local positive force feedback (F+) during the stance period of gait (FIG. 2B). This force reflex is modeled in the same way as in Geyer, H., Seyfarth, A., Blickhan, R., 2003. Positive force feedback in bouncing gaits? Proc. R. Soc. Lond. B 270, 2173-2183. Under positive force feedback, the stimulation Sm(t) of a muscle m is the sum of a pre-stimulation S0,m, and the muscle's time-delayed (Δt) and gained (G) force Fm: Sm(t)=S0,m+GmFm(t−Δtm).

While compliant leg behavior is essential, it also threatens joint stability in segmented legs [Seyfarth, A., Günther, M., Blickhan, R., 2001. Stable operation of an elastic three-segmented leg. Biol. Cybern. 84, 365-382; Günther, M., Keppler, V., Seyfarth, A., Blickhan, R., 2004. Human leg design: optimal axial alignment under constraints. J. Math. Biol. 48, 623-646]. In segmented legs, the knee and ankle torques, τ_(k) and τ_(a), obey the static equilibrium τ_(k)/τ_(a)=h_(k)/h_(a), where h_(k) and h_(a) are the perpendicular distances from the knee and the ankle to the leg force vector Fleg, respectively. In effect, a large extension torque at one joint forces the other joint closer to Fleg, threatening its overextension for spring-like behaving legs [for details see Seyfarth, A., Günther, M., Blickhan, R., 2001. Stable operation of an elastic three-segmented leg. Biol. Cybern. 84, 365-382].

This tendency to overextend at the knee or the ankle is countered by adding the gastrocnemius (GAS) 245 and tibialis anterior (TA) 250 muscles (FIG. 2C). Like SOL and VAS, the biarticular GAS uses local positive force feedback (F+) during the stance period of gait. This muscle reflex not only prevents knee hyperextension resulting from large extension torques at the ankle, but also contributes to generating an overall compliant leg behavior. In contrast, the monoarticular TA uses local positive length feedback (L+) with S_(TA)(t)=S_(0,TA)+G_(TA)(l_(CE,TA)−l_(off,TA))(t−Δ_(t,TA)) where l_(CE,TA) is the TA fiber length and l_(off,TA) is a length offset. Flexing the foot, TA's L+ prevents the ankle from overextending when large knee torques develop. This muscle reflex is not required however if sufficient activity of the ankle extensor muscles preserves the torque equilibrium of knee and ankle. To avoid that TA unnecessarily fights SOL in this situation, the TA stimulation is inhibited with a negative force feedback (F−) from the SOL, resulting in S_(TA)(t)=S_(0,TA)+G_(TA)(l_(CE,TA)−l_(off,TA))(t−Δ_(t,TA))−G_(SOLTA)F_(SOL)(t−Δt_(SOL)). To further protect the knee from hyperextending, the VAS gets inhibited if the knee extends beyond a 170 deg threshold, S_(VAS)(t)=S_(0,VAS)+G_(VAS)F_(VAS)(t−Δt_(VAS))−k_(φ)Δφ_(k)(t−Δt_(k)), where k_(φ) is a proportional gain, Δφ_(k)=φ_(k)−170 deg, and φ_(k) is the knee angle. This reflex inhibition is only active if Δφ>0 and the knee is actually extending.

Upper body and its balance. In the next step of evolving from the conceptual spring-mass model into a neuromuscular biped, the point mass representation is discarded and an upper body 255 around which the legs can be swung (FIG. 2D) is introduced. This upper body 255 combines head, arms and trunk (HAT). To balance the HAT 255 during locomotion, to each leg is added a gluteus muscle group (GLU) 260 and a hip flexor muscle group (HFL) 265. The GLU 260 and the HFL 265 are stimulated with a proportional-derivative signal of the HAT's 255 forward lean angle θ with respect to gravity, S_(GLU/HFL)˜±[k_(p)(θ−θ_(ref))+k_(d)dθ/dt], where k_(p) and k_(d) are the proportional and derivative gains, and θ_(ref) is a reference lean angle [for similar approaches compare, for instance, Günther, M., Ruder, H., 2003. Synthesis of two-dimensional human walking: a test of the λ-model. Biol. Cybern. 89, 89-106]. Also included is the biarticular hamstring muscle group (HAM) 270 with S_(HAM)˜S_(GLU) to counter knee hyperextension that results from a large hip torque developed by the GLU 260 when pulling back the heavy HAT 255. Since hip torques can only balance the HAT 255 if the legs bear sufficient weight, the stimulations of the GLU 260, HAM 270, and HFL 265 are modulated for each leg proportionally to the amount of body weight it bears. As a result, each leg's hip muscles contribute to the HAT's balance control only during stance.

Swing leg pro- and retraction. The human model's structure is complete, except for a muscle-reflex control that produces swing leg pro- and retraction. It is assumed that a stance leg's functional importance reduces in proportion to the amount of body weight (bw) borne by the contralateral leg, and initiate swing leg protraction already in double support (FIG. 2E). The human model detects which leg enters stance last (contralateral leg), and suppresses F+ of the ipsilateral leg's VAS 240 in proportion to the weight the contralateral leg bears, S_(VAS)(t)=S_(0,VAS)+G_(VAS)F_(VAS)(t−Δt_(VAS))−k_(φ)Δφ_(k)(t−Δt_(k))−k_(bw)|F_(leg) ^(contra)|. The contralateral suppression allows the knee to break its functional spring behavior, and flex while the ankle extends, pushing the leg off the ground and forward. While this catapult mechanism can initiate swing only if the ankle pushes sufficiently, the model further prepares swing leg protraction by increasing the stimulation of the HFL 265, and decreasing that of the GLU 260, by a fixed amount ΔS in double support.

During actual swing, the main reliance is on a leg's ballistic motion, but it is influenced in two ways (FIG. 2F). On one hand, protraction of the swing leg is facilitated. The HFL 265 is stimulated using positive length feedback (L+) biased by the forward pitch angle θ_(ref) of the HAT 255 at the stance-to-swing transition, S_(HFL)(t)=S_(0,HFL)+k_(lean)(θ−θ_(ref))_(TO)+G_(HFL)(l_(CE,HFL)−l_(off,HFL))(t−Δ_(t,HFL)). Using this approach, it is ensured that the swing leg's ballistic motion gains the momentum to bring it forward in time [Mochon, S., McMahon, T., 1980. Ballistic walking. J. Biomech. 13 (1), 49-57].

Furthermore, the swing leg is also prevented from overreaching and its retraction is ensured. If legs reach and maintain a proper orientation during swing, legged systems self-stabilize into a gait cycle [McGeer, T., 1990. Passive dynamic walking. Int. J. Rob. Res. 9 (2), 62-82; Seyfarth, A., Geyer, H., Günther, M., Blickhan, R., 2002. A movement criterion for running. J. of Biomech. 35, 649-655; Ghigliazza, R., Altendorfer, R., Holmes, P., Koditschek, D., 2003. A simply stabilized running model. SIAM J. Applied. Dynamical Systems 2 (2), 187-218; Geyer, H., Seyfarth, A., Blickhan, R., 2006. Compliant leg behaviour explains the basic dynamics of walking and running. Proc. R. Soc. Lond. B 273, 2861-2867]. The tolerance of this mechanical self-stability against disturbances can largely be enhanced if swing legs additionally retract before landing [Seyfarth, A., Geyer, H., 2002. Natural control of spring-like running—optimized self-stabilization. In: Proceedings of the 5th international conference on climbing and walking robots. Professional Engineering Publishing Limited, pp. 81-85; Seyfarth, A., Geyer, H., Herr, H. M., 2003. Swing-leg retraction: a simple control model for stable running. J. Exp. Biol. 206, 2547-2555]. To implement this halt-and-retract strategy, three muscle reflexes are included in the human model. The overreaching of the swing leg that would result from the forward impulse the leg receives when the knee reaches full extension during protraction is prevented. Hereto, the HFL's L+ is inhibited proportional to the stretch which the HAM receives in swing, S_(HFL)(t)=k_(lean)(θ−θ_(ref))TO+G_(HFL)(l_(CE,HFL)−l_(off,HFL))(t−Δ_(t,HFL))−G_(HAMHFL)(l_(CE,HAM)−l_(off,HAM))(t−Δ_(t,HAM)). In addition, F+ is used for the GLU, S_(GLU)(t)=S_(0,GLU)+G_(GLU)F_(GLU)(t−Δt_(GLU)), and for the HAM, S_(HAM)(t)=S_(0,HAM)+G_(HAM)F_(HAM)(t−Δt_(HAM)), to ensure that, dependent on the actual protraction momentum, the swing leg not only halts, but also transfers part of this momentum into leg straightening and retraction. Finally, the TA L+ introduced to ensure foot clearance is kept throughout the swing. The SOL, GAS, and VAS remain silent during this phase.

Reflex control parameters. The different reflex contributions to the muscle stimulations Sm(t) are governed through the equations used in the model. No parameter optimization was performed. Parameters were derived from previous knowledge of reflex behavior (F+, L+) or by making plausible estimates. All muscle stimulations are limited in range from 0.01 to 1 before being translated into muscle activations A_(m)(t). Table 1 presents the stance reflex equations used in the preferred embodiment.

TABLE 1 $\begin{matrix} {{S_{SOL}(t)} = {S_{0,{SOL}} + {G_{SOL}{F_{SOL}\left( t_{1} \right)}}}} \\ {= {0.01 + {{1.2/F_{\max,{SOL}}}{F_{SOL}\left( t_{1} \right)}}}} \end{matrix}\quad$ $\begin{matrix} {\left. {{S_{TA}(t)} = {S_{0,{TA}} + {G_{TA}\left\lbrack {{l_{{CE},{TA}}\left( t_{1} \right)} - l_{{off},{TA}}} \right)}}} \right\rbrack - {G_{{SOL},{TA}}{F_{SOL}\left( t_{1} \right)}}} \\ {\left. \left. {= {0.01 + {1.1\left\lbrack {{l_{{CE},{TA}}\left( t_{1} \right)} - {0.71l_{{opt},{TA}}}} \right.}}} \right) \right\rbrack -} \\ {{{0.3/F_{\max,{SOL}}}{F_{SOL}\left( t_{1} \right)}}} \end{matrix}\quad$ $\begin{matrix} {{S_{GAS}(t)} = {S_{0,{GAS}} + {G_{GAS}{F_{GAS}\left( t_{1} \right)}}}} \\ {= {0.01 + {{1.1/F_{\max,{GAS}}}{F_{GAS}\left( t_{1} \right)}}}} \end{matrix}\quad$ $\begin{matrix} {{S_{VAS}(t)} = {S_{0,{VAS}} + {G_{VAS}{F_{VAS}\left( t_{m} \right)}} - {{k_{\phi}\left\lbrack {{\phi_{k}\left( t_{m} \right)} - \phi_{k,{off}}} \right\rbrack}\left\lbrack {{\phi_{k}\left( t_{m} \right)} > \phi_{k,{off}}} \right\rbrack}}} \\ {{\left\lbrack {{d\; {\phi_{k}/{{dt}\left( t_{m} \right)}}} > 0} \right\rbrack - {k_{bw}{{F_{leg}^{contra}\left( t_{s} \right)}}{\,^{*}{DSup}}}}} \\ {= {0.09 + {{1.15/F_{\max,{VAS}}}{F_{VAS}\left( t_{m} \right)}} - {1.15\left\lbrack {{\phi_{k}\left( t_{m} \right)} - 2.97} \right\rbrack}}} \\ {{{\left\lbrack {{\phi_{k}\left( t_{m} \right)} > 2.97} \right\rbrack \left\lbrack {{d\; {\phi_{k}/{{dt}\left( t_{m} \right)}}} > 0} \right\rbrack} - {0.00167\; {{F_{leg}^{contra}\left( t_{s} \right)}}{\,^{*}{DSup}}}}} \end{matrix}\quad$ $\begin{matrix} {{S_{HAM}(t)} = {S_{0,{HAM}} + \left\{ {{k_{p}\left\lbrack {{\theta \left( t_{s} \right)} - \theta_{ref}} \right\rbrack} + {k_{d}d\; {\theta/{{dt}\left( t_{s} \right)}}}} \right\} + {k_{bw}{{F_{leg}^{ipsi}\left( t_{s} \right)}}\,}}} \\ {= {0.05 + \left\{ {{1.9\left\lbrack {{\theta \left( t_{s} \right)} - 0.105} \right\rbrack} + {0.25d\; {\theta/{{dt}\left( t_{s} \right)}}}} \right\} + {0.00167{{F_{leg}^{ipsi}\left( t_{s} \right)}}}}} \end{matrix}\quad$ $\begin{matrix} {{S_{GLU}(t)} = {S_{0,{GLU}} + \left\{ {{k_{p}\left\lbrack {{\theta \left( t_{s} \right)} - \theta_{ref}} \right\rbrack} + {k_{d}d\; {\theta/{{dt}\left( t_{s} \right)}}}} \right\} + {k_{bw}{{F_{leg}^{ipsi}\left( t_{s} \right)}}} - \,}} \\ {{{\Delta S}{\,^{*}{DSup}}}} \\ {= {0.05 + \left\{ {{1.3\left\lbrack {{\theta \left( t_{s} \right)} - 0.105} \right\rbrack} + {0.25d\; {\theta/{{dt}\left( t_{s} \right)}}}} \right\} +}} \\ {{{0.00167\; {{F_{leg}^{ipsi}\left( t_{s} \right)}}} - {0.25{\,^{*}{DSup}}}}} \end{matrix}\quad$ $\begin{matrix} {{S_{HFL}(t)} = {S_{0,{HFL}} + \left\{ {{k_{p}\left\lbrack {{\theta \left( t_{s} \right)} - \theta_{ref}} \right\rbrack} + {k_{d}d\; {\theta/{{dt}\left( t_{s} \right)}}}} \right\} - {k_{bw}{{F_{leg}^{ipsi}\left( t_{s} \right)}}} + \,}} \\ {{{\Delta S}{\,^{*}{DSup}}}} \\ {= {0.05 + \left\{ {{1.9\left\lbrack {{\theta \left( t_{s} \right)} - 0.105} \right\rbrack} + {0.25d\; {\theta/{{dt}\left( t_{s} \right)}}}} \right\} -}} \\ {{{0.00167\; {{F_{leg}^{ipsi}\left( t_{s} \right)}}} + {0.25{\,^{*}{DSup}}}}} \end{matrix}\quad$ (t₁=t−20 ms, t_(m)=t−10 ms, and t_(s)=t−5 ms, DSup is 1 if leg is trailing leg in double support, otherwise 0, { }_(+/−) refers to only positive/negative values)

Table 2 presents the swing reflex equations used in the preferred embodiment.

TABLE 2   $\begin{matrix} {{S_{SOL}(t)} = S_{0,{SOL}}} \\ {= 0.01} \end{matrix}\quad$ $\begin{matrix} \left. {{S_{TA}(t)} = {S_{0,{TA}} + {G_{TA}\left\lbrack {{l_{{CE},{TA}}\left( t_{1} \right)} - l_{{off},{TA}}} \right)}}} \right\rbrack \\ \left. {= {0.01 + {1.1\left\lbrack {{l_{{CE},{TA}}\left( t_{1} \right)} - {0.71l_{{opt},{TA}}}} \right)}}} \right\rbrack \end{matrix}\quad$ $\begin{matrix} {{S_{GAS}(t)} = S_{0,{GAS}}} \\ {= 0.01} \end{matrix}\quad$ $\begin{matrix} {{S_{VAS}(t)} = S_{0,{VAS}}} \\ {= 0.01} \end{matrix}\quad$ $\begin{matrix} {{S_{HAM}(t)} = {S_{0,{HAM}} + {G_{HAM}{F_{HAM}\left( t_{s} \right)}}}} \\ {= {0.01 + {{0.65/F_{\max,{HAM}}}{F_{HAM}\left( t_{s} \right)}}}} \end{matrix}\quad$ $\begin{matrix} {{S_{GLU}(t)} = {S_{0,{GLU}} + {G_{GLU}{F_{GLU}\left( t_{s} \right)}}}} \\ {= {0.01 + {{0.4/F_{\max,{GLU}}}{F_{GLU}\left( t_{s} \right)}}}} \end{matrix}\quad$ $\begin{matrix} {{S_{HFL}(t)} = {S_{0,{HFL}} + {{GH}_{FL}\left\lbrack {{l_{{CE},{HFL}}\left( t_{s} \right)} - l_{{off},{HFL}}} \right\rbrack} - \,}} \\ {{{G_{{HAM},{HFL}}\left\lbrack {{l_{{CE},{HAM}}\left( t_{s} \right)} - l_{{off},{HAM}}} \right\rbrack} + {\left\{ {k_{lean}\left\lbrack {{\theta \left( t_{s} \right)} - \theta_{ref}} \right\rbrack} \right\} {PTO}}}} \\ {= {0.01 + {0.35\left\lbrack {{l_{{CE},{HFL}}\left( t_{s} \right)} - {0.6l_{{opt},{HFL}}}} \right\rbrack} -}} \\ {{{4\left\lbrack {{l_{{CE},{HAM}}\left( t_{s} \right)} - {0.85l_{{opt},{HAM}}}} \right\rbrack} + {\left\{ {1.15\left\lbrack {{\theta \left( t_{s} \right)} - 0.105} \right\rbrack} \right\} {PTO}}}} \end{matrix}\quad$ ({ }PTO: constant value taken at previous take off.)

Results. Although the human model has no central pattern generator (CPG) that feed-forwardly activates its muscles, it switches for each leg between the different reflexes for stance and swing using sensors located at the ball and heel of each foot to detect ground. As a result, the model's dynamic interaction with its mechanical environment becomes a vital part of generating muscle activities. FIG. 3 graphically depicts pattern generation according to this aspect of the invention. In FIG. 3, instead of a central pattern, reflexes generate the muscle stimulations, S_(m) 305, 310. Left (L) 320 and right (R) 330 leg have separate stance 340, 345 and swing 350, 355 reflexes, which are selected based on contact sensing 360, 365 from ball and heel sensors 370, 375. The reflex outputs depend on mechanical inputs, M_(i) 380, 385, intertwining mechanics and motor control.

Walking gait. To study how important this interdependence of mechanics and motor control can be to human locomotion, the model was started with its left leg in stance and its right leg in swing at a normal walking speed v0=1.3 ms−1. Since the modeled muscle reflexes include time delays of up to 20 ms, all muscles are silent at first. FIGS. 4A and 4B depict walking of a human model self-organized from dynamic interplay between model and ground and the corresponding ground reaction force, respectively, according to one aspect of the present invention. In FIGS. 4A and 4B, snapshots of human model taken every 250 ms (FIG. 4A) and corresponding model GRF (FIG. 4B) are shown, with separate plots for left 405, 410 and right 415,420 legs (30 Hz low-pass filtered). Starting with a horizontal speed of 1.3 ms⁻¹, the model slows down in the first two steps, but then rapidly recovers into walking at the same speed. Leg muscles are shown only for the right leg 415, indicating muscle activation>10%. Initial conditions for φ_(a,k,h) (definition of ankle, knee and hip angle) for each leg were: φ_(a,k,h)=85 deg, 175 deg, 175 deg (left leg) and φ_(a,k,h)=90 deg, 175 deg, 140 deg (right leg).

Because of these disturbed initial conditions, the model slightly collapses and slows down in its first step (FIG. 4A). If its parameters are chosen properly, however, the model rapidly recovers in the following steps, and walking self-organizes from the dynamic interplay between model and ground. Here the vertical ground reaction force (GRF) of the legs in stance shows the M-shape pattern characteristic for walking gaits (FIG. 4B), indicating similar whole-body dynamics of model and humans for steady state walking.

Steady-state patterns of angles, torques and muscle activations. This similarity also holds upon closer inspection; the model shows qualitative agreement with angle, torque and muscle activation patterns known from human walking data. FIGS. 5A-C compare steady state walking at 1.3 ms⁻¹ for the model and a human subject for hip (FIG. 5A), knee (FIG. 5B), and ankle (FIG. 5C), respectively, according to one aspect of the present invention. In FIGS. 5A-C, normalized to one stride from heel-strike to heel-strike of the same leg, the model's steady-state patterns of muscle activations, torques, and angles of hip, knee and ankle are compared to human walking data (adapted from Perry, 1992). Vertical dotted lines 510 around 60% of stride indicate toe off. Compared muscles are adductor longus (HFL) 520, upper gluteus maximum (GLU) 530, semimembranosis (HAM) 540, and vastus lateralis (VAS) 550.

The strongest agreement between model prediction and walking data can be found at the ankle (FIG. 5C). The reflex model not only generates ankle kinematics φ_(a) and torques τ_(a) observed for the human ankle in walking, but also predicts SOL, TA and GAS activities that resemble the experimental SOL, TA and GAS activities as inferred from their surface electromyographs. For SOL and GAS, this activity is generated exclusively by their local F+ reflexes in stance. For TA, its L+ reflex responds with higher activity to plantar flexion of the foot in early stance, but gets suppressed by F− from SOL during the remainder of that phase. Only when SOL activity reduces at the transition from stance to swing (60% of stride), does the TA's L+ resume, pulling the foot against plantar flexion.

The comparison shows a weaker agreement for the knee and the hip. For instance, although the general trajectory φ_(k) of the human knee is captured by the model, its knee flexes about 10 degree or 30% more than the human's in early stance (FIG. 5B). Related to this larger knee flexion, the model lacks the observed VAS activity in late swing that continues into early stance. Only after heel-strike, the F+ of VAS engages and can activate the muscle group in response to the loading of the leg. The delay in extensor activities causes not only a relatively weak knee in early stance, but also the heavy HAT to tilt forward after impact. Since the balance control of the HAT engages gradually with the weight borne by the stance leg, the balance reflexes are silent until heel-strike and then must produce unnaturally large GLU and HAM activities to return the HAT to its reference lean (FIG. 5C). Hence, the model's hip trajectory φ_(h) and torque pattern τ_(h) least resemble that of humans whose hip extensors GLU and HAM are active before impact and can prevent such an exaggerated tilt of the trunk.

Self-adaptation to ground changes. Despite its limited reflex control, the human model tolerates sudden, and self-adapts to permanent, changes of the ground level. FIGS. 6A-D show an example in which the model encounters a sequence of stairs going up 4 cm each. FIGS. 6A-D depict adaptation to walking up stairs, including snapshots of the model (FIG. 6A), net work (FIG. 6B), extensor muscle activation patterns (FIG. 6C), and the corresponding ground reaction force (FIG. 6D), according to one aspect of the present invention. In FIGS. 6A-D, approaching from steady-state walking at 1.3 ms⁻¹, eight strides of the human model are shown covering five steps of 4 cm incline each. The model returns to steady-state walking on the 8th stride. One stride is defined from heel-strike to heel-strike of the right leg. Shown in FIG. 6A are snapshots of the model at heel-strike and toe-off of the right leg. For this leg are further shown, in FIG. 6B, the net work during stance generated at hip, knee and ankle with positive work being extension work; in FIG. 6C, the activation patterns of the five extensor muscles of each stride; and, in FIG. 6D, the corresponding ground reaction forces 650 normalized to body weight (bw), with ground reaction forces of the left leg 660 are included for comparison.

Approaching from steady-state walking (1st stride), the model hits the stairs at the end of the 2nd stride with the foot of its outstretched right leg (FIG. 6A). This early impact slows down the model and tilts the upper body forward, which is countered by a large hip torque generated by the GLU and HAM (3rd stride, FIGS. 6B and 6C). Since hip extension torques tend to also extend the knee, the VAS does not feel as much force as in steady-state and its force feedback control lowers its muscle stimulation (FIG. 6C), even though the net work at the knee during stance remains about the same as in steady state. In contrast, the slow down of the model reduces the force the ankle extensors GAS and SOL feel during stance, and their force feedback reflexes produce slightly less muscle stimulation, lowering the net work of the ankle (FIGS. 6B and 6C). In strides 4 and 5 the model settles into upstair walking at about 1 ms⁻¹ where the forward and upward thrust is generated mainly at the hip and knee. After reaching the plateau in the 6th stride, the model recovers into its original steady-state walking speed of 1.3 ms⁻¹ in the 8th stride.

FIGS. 7A-D depict adaptation to walking down stairs, including snapshots of the model (FIG. 7A), net work (FIG. 7B), extensor muscle activation patterns (FIG. 7C), and the corresponding ground reaction force (FIG. 7D), according to one aspect of the present invention. In FIGS. 7A-D, approaching from steady-state walking at 1.3 ms⁻¹, eight strides of the human model are shown covering five steps of 4 cm incline each. The model returns to steady-state walking on the 8th stride. One stride is defined from heel-strike to heel-strike of the right leg. Shown in FIG. 7A are snapshots of the model at heel-strike and toe-off of the right leg. For this leg are further shown, in FIG. 7B, the net work during stance generated at hip, knee and ankle with positive work being extension work; in FIG. 7C, the activation patterns of the five extensor muscles of each stride; and, in FIG. 7D, the corresponding ground reaction forces 750 normalized to body weight (bw), with ground reaction forces of the left leg 760 are included for comparison. The model returns to steady state walking at 1.3 ms⁻¹ in the 14th stride after covering five steps down with 4 cm decline each.

FIGS. 7A-D continues the walking sequence with the model encountering stairs going down. At the end of the 9th stride, the model hits the first step down with its right foot (FIG. 7A). The downward motion accelerates the model and results in an overall larger first impact of the right leg in the 10th stride with a stronger response of most extensor muscles (FIGS. 7C and 7D). Only the GAS generates less force, because the knee stays more flexed than usual in this stride. As a result, positive net work at the ankle increases substantially (FIG. 7B). This increase and a larger HFL stimulation (not shown) caused by the forward lean of the upper body at its take-off (FIG. 7A) propel the right leg forward in swing increasing the step length (FIG. 7A). After the transitional 10th stride, the model keeps the larger step length in the downward motion (strides 11 and 12), where the model's downward acceleration is countered by increased activity of the GLU, HAM and VAS immediately following impact (FIGS. 7C and 7D), which reduces net positive work at the hip and increases net negative work at the knee (FIG. 7B), and stabilizes the model into walking down at about 1.5 ms⁻¹. Back on level ground, the lack of downward acceleration slows down the model, which automatically reduces its step length (FIG. 7A) and drives it back into steady-state walking at 1.3 ms⁻¹ within the 13th and 14th step.

For both walking up and down stairs, no single control is responsible. The key to the model's tolerance and adaptation are its dynamic muscle-reflex responses. The rebound of the stance leg depends on how much load the leg extensors SOL, GAS and VAS feel, which guarantees that the leg yields sufficiently to allow forward progression when going up, but brakes substantially when going down. On the other hand, the forward propulsion of the swing leg varies with the model dynamics. Sudden deceleration after impact of the opposite leg, forward lean of the upper body, and ankle extension rate near the end of stance—all contribute to leg propulsion in swing. These combined features ensure that the swing leg protracts enough in upstair walking and substantially in downstair walking. For the latter, the force feedbacks of GLU and HAM constrain excess rotations of the leg and instead force it to rapidly retract and straighten.

Muscle tendon units. All 14 muscle-tendon units (MTUs) of the biped have the same model structure. FIG. 8 is a schematic of a muscle-tendon model, according to one aspect of the present invention. In FIG. 8, active, contractile element (CE) 810 together with series elasticity (SE) 820 form the muscle-tendon unit (MTU) in normal operation. If CE 810 stretches beyond its optimum length l_(CE) 830 (l_(CE)>l_(opt) 840), parallel elasticity (PE) 850 engages. Conversely, buffer elasticity (BE) 860 prevents the active CE 810 from collapsing if SE 820 is slack (l_(MTU) 870−l_(CE) 830<l_(slack) 880).

As seen in FIG. 8, an active, Hill-type contractile element (CE) produces force in line with a series elasticity (SE). Although the MTUs are fitted into the skeleton such that the individual CEs operate mainly on the ascending limb of their force-length relationship, the MTU model includes a parallel elasticity (PE), which engages if the CE stretches beyond its optimum length l_(opt). In addition, a buffer elasticity (BE) ensures that the CE cannot collapse when the geometry of the leg shortens the MTU so much that it becomes slack. Note that BE is merely a numerical tool that allows the MTU to describe a slack muscle, for instance, a slack GAS when the knee overly flexes. BE does however not result in forces outside the MTU.

Table 3 presents individual MTU parameters. All parameters are estimated from Yamaguchi et al. [Yamaguchi, G. T., Sawa, A. G.-U., Moran, D. W., Fessler, M. J., Winters, J. M., 1990. A survey of human musculotendon actuator parameters. In: Winters, J., Woo, S.-Y. (Eds.), Multiple Muscle Systems: Biomechanics and Movement Organization. Springer-Verlag, New York, pp. 717-778]. The maximum isometric forces F_(max) are estimated from individual or grouped muscle-physiological cross-sectional areas assuming a force of 25N per cm⁻². The maximum contraction speeds v_(max) are set to 6l_(opt) s⁻¹ for slow muscles and to 12l_(opt) s⁻¹ for medium fast muscles. The optimum CE lengths l_(opt) and the SE slack lengths l_(slack) reflect muscle fiber and tendon lengths.

TABLE 3 SOL TA GAS VAS HAM GLU HFL F_(max) (N) 4000 800 1500 6000 3000 1500 2000 V_(max) (l_(opt) s⁻¹) 6 12 12 12 12 12 12 l_(opt) (cm) 4 6 5 8 10 11 11 l_(slack) (cm) 26 24 40 23 31 13 10

Details on how CE and SE were modeled can be found in Geyer et al. [Geyer, H., Seyfarth, A., Blickhan, R., 2003. Positive force feedback in bouncing gaits? Proc. R. Soc. Lond. B 270, 2173-2183]. The force of the CE, F_(CE)=A F_(max)f_(l)(l_(CE))f_(v)(v_(CE)), is i a product of muscle activation A, CE force-length relationship f_(l)(l_(CE)), and CE force-velocity relationship f_(v)(v_(CE)). Based on this product approach, the MTU dynamics are computed by integrating the CE velocity v_(CE), which is found by inverting f_(v)(v_(CE)). Given that F_(SE)=F_(CE)+FP_(E)−F_(BE), f_(v)(v_(CE))=(F_(SE)−F_(PE)+F_(BE))/(A F_(max)f_(l)(l_(CE))). This equation has a numerically critical point during muscle stretch when F_(SE)−F_(PE) approaches zero. To speed up simulations, this critical point is avoided by introducing f_(v)(v_(CE)) into the force production of the parallel elasticity F_(PE)˜(l_(CE)−l_(opt))²f_(v)(v_(CE)). Note that PE engages outside the normal range of operation in the model, and like BE, plays a minor role for the muscle dynamics during normal locomotion. With this approach, however, f_(v)(v_(CE))=(F_(SE)+F_(BE))/(A F_(max)f_(l)(l_(CE))+F_(PE)) is obtained, which can numerically be integrated using coarse time steps. While this approach is convenient to speed up the model simulation, it was also critical when muscle dynamics were emulated on PC boards with fixed and limited time resolution.

The MTUs have common and individual parameters. The common parameters include the time constant of the excitation contraction coupling, t_(ecc)=0.01; the CE force-length relationship's width, w=0.56l_(opt), and residual force factor, c=0.05; the CE force-velocity relationship's eccentric force enhancement, N=1.5, and shape factor, K=5; and the SE reference strain, ε_(ref)=0.04 [for details, see Geyer, H., Seyfarth, A., Blickhan, R., 2003. Positive force feedback in bouncing gaits? Proc. R. Soc. Lond. B 270, 2173-2183]. Also common parameters are the PE reference strain ε_(PE)=w where F_(PE)=F_(max)(l_(CE)/l_(opt)−1)²/ε_(PE) ²f_(v)(v_(CE)), and the BE rest length l_(min)=l_(opt)−w and its reference compression ε_(BE)=w/2 where F_(BE)=F_(max)[(l_(min)−l_(CE))/l_(opt)]²/ε_(PE) ². The individual MTU attachment parameters are readily available from the literature and distinguish each muscle or muscle group. Their values are listed in Table 4.

TABLE 4 MTU attachment parameters ankle knee hip SOL TA GAS GAS VAS HAM HAM GLU HFL r₀ 5 4 5 5 6 5 8 10 10 (cm) φ_(max) 110 80 110 140 165 180 — — — (deg) φ_(ref) 80 110 80 165 125 180 155 150 180 (deg) ρ 0.5 0.7 0.7 0.7 0.7 0.7 0.7 0.5 0.5

Musculoskeletal connections and mass distribution. The MTUs connect to the skeleton by spanning one or two joints. The transfer from muscle forces F_(m) to joint torques τ_(m) is modeled using variable lever arms r_(m)(φ)=r₀ cos(φ−φ_(max)) for the ankle and knee where φ is the joint angle, φ_(max) is the angle at which r_(m) reaches its maximum, and τ_(m)=r_(m)(φ)F_(m). For the hip, it is simply assumed that r_(m)(φ)=r₀. On the other hand, changes Δl_(m) in MTU lengths are modeled as Δl_(m)=ρr[sin(φ−φ_(max)−sin(φ_(ref)−φ_(max))] for the ankle and knee; and as Δl_(m)=ρr(φ−φ_(ref)) for the hip. The reference angle φ_(ref) is the joint angle where l_(m)=l_(opt)+l_(slack). The factor ρ accounts for muscle pennation angles and ensures that an MTU's fiber length stays within physiological limits throughout the working range of the joint. The specific parameters for each muscle and joint are listed in Table 4. These values are either supported by experimental evidence [Muraoka, T., Kawakami, Y., Tachi, M., Fukunaga, T., 2001. Muscle fiber and tendon length changes in the human vastus lateralis during slow pedaling. J. Appl. Physiol. 91, 2035-2040; Maganaris, C., 2001. Force-length characteristics of in vivo human skeletal muscle. Acta Physiol. Scand. 172, 279-285; Maganaris, C., 2003. Force-length characteristics of the in vivo human gastrocnemius muscle. Clin. Anat. 16, 215-223; Oda, T., Kanehisa, H., Chino, K., Kurihara, T., Nagayoshi, T., Kato, E., Fukunaga, T., Kawakami, Y., 2005. In vivo length-force relationships on muscle fiver and muscle tendon complex in the tibialis anterior muscle. Int. J. Sport and Health Sciences 3, 245-252], or were obtained through rough anatomical estimates.

The seven segments of the human model are simple rigid bodies whose parameters are listed in Table 5. Their values are similar to those used in other modeling studies, for instance, in Günther and Ruder [Günther, M., Ruder, H., 2003. Synthesis of two-dimensional human walking: a test of the λ-model. Biol. Cybern. 89, 89-106]. The segments are connected by revolute joints. As in humans, these joints have free ranges of operation (70°<φ_(a)<130°, φ_(k)<175° and φ_(h)<230°) outside of which mechanical soft limits engage, which is modeled in the same way as the ground impact points. The model's segments have different masses m_(S) and lengths l_(S), and characteristic distances of their local center of mass, d_(G,S), and joint location, d_(J,S) (measured from distal end), and inertias Θ_(S).

TABLE 5 Feet Shanks Thighs HAT l_(S) (cm) 20 50 50 80 d_(G,S) (cm) 14 30 30 35 d_(J,S) (cm) 16 50 50 — m_(S) (cm) 1.25 3.5 8.5 53.5 Θ_(S) (kgm2) 0.005 0.05 0.15 3

Ground contacts and joint limits. Each foot segment of the bipedal model has contact points at its toe and heel. When impacting the ground, a contact point (CP) gets pushed back by a vertical reaction force F_(y)=−F_(ref)f_(l)f_(v), which, like the muscle force, is the product of a force-length relationship f_(l)(Δy_(CP))=Δy_(CP)/Δy_(ref) and a force-velocity relationship f_(v)(dy_(CP)/dt)=1−dy_(CP)/dt/v_(max) (FIG. 9). This product approach to modeling vertical reaction forces is similar to existing approaches that describe the vertical force as the sum of a spring and a nonlinear spring-damper term [Scott, S., Winter, D., 1993. Biomechanical model of the human foot: kinematics and kinetics during the stance phase of walking. J. Biomech. 26 (9), 1091-1104; Gerritsen, K., van den Bogert, A., Nigg, B., 1995. Direct dynamics simulation of the impact phase in heel-toe running. J. Biomech. 28 (6), 661-668; Günther, M., Ruder, H., 2003. Synthesis of two-dimensional human walking: a test of the λ-model. Biol. Cybern. 89, 89-106]. By separating spring and damper terms, however, the parameters of the contact model can be interpreted as two basic material properties: a ground stiffness k=F_(ref)/Δy_(ref) and a maximum relaxation speed v_(max), which characterizes how quickly the ground surface can restore its shape after being deformed. For instance, v_(max)=∞ describes a perfectly elastic ground impact where the ground always pushes back against the CP, and v_(max)=0 describes a perfectly inelastic impact where the ground, like sand, pushes back on the CP for downward velocities, but cannot push back for upward velocities. Note that the same impact model is used to describe the mechanical soft limits of the model's joints (see previous section) with a soft limit stiffness of 0.3N m deg⁻¹ and a maximum relaxation speed of 1 deg s⁻¹.

FIGS. 9A-C depict a contact model, according to one aspect of the present invention. In FIGS. 9A-C, contact occurs 910 if contact point 920 falls below y₀. The vertical ground reaction force F_(y) is, like the muscle force, modeled as the product of a force-length (f_(l)) and a force-velocity relationship (f_(v)) with Δy_(ref) being the ground compression at which F_(y)=F_(ref) when dy/dt=0, and dy_(ref)/dt being the maximum relaxation speed of the ground (small diagrams). Initially, the horizontal ground reaction force F_(x) is modeled as sliding friction proportional to Fy with sliding coefficient μ_(sl). If however contact point 920 slows down 930 to below a minimum speed v_(lim), the horizontal model switches to stiction 930. During stiction 930, F_(x) is also modeled as the product of force-length and force-velocity relationships, which slightly differ from those earlier in order to allow for interactions with the ground in both directions around the stiction reference point x₀. The model switches back to sliding friction if F_(x) exceeds the stiction limit force μ_(st) F_(y). Parameters: F_(ref)=815N, Δy_(ref)=0.01 m, dy_(ref)/dt=0.03 ms⁻¹, Δx_(ref)=0.1 m, dx_(ref)/dt=0.03 ms⁻¹, v_(lim)=0.01 ms⁻¹, μ_(sl)=0.8, μ_(st)=0.9.

In addition to the vertical reaction force, a horizontal reaction force is applied to the CP during ground contact. Initially, this force is modeled as a kinetic friction force that opposes the CP's motion on the ground with a force F_(x)=μ_(sl) F_(y). When the CP slows down to below a speed v_(lim), the horizontal reaction force is modeled as a stiction force computed in a manner similar to that in which the vertical impact force is computed (FIGS. 9A-C). Stiction changes back to kinetic friction if the stiction force exceeds a limit force F_(lim)=μ_(st) F_(y). Thus, dependent on the transition conditions, both types of horizontal reaction force interchange until the CP leaves the ground surface.

The results suggest that mechanics and motor control cannot be viewed separately in human locomotion. A neuromuscular model of human locomotion according to one aspect of the invention self-organizes into the walking gait after an initial push, tolerates sudden changes in ground level, and adapts to stair walking without interventions. Central to this model's tolerance and adaptiveness is its reliance on muscle reflexes, which integrate sensory information about locomotion mechanics into the activation of the leg muscles. Having no CPG, the model shows that in principle no central input is required to generate walking motions, suggesting that reflex inputs that continuously mediate between the nervous system and its mechanical environment may even take precedence over central inputs in the control of normal human locomotion.

In addition, the model results suggest that these continuous reflex inputs encode principles of legged mechanics. Current experimental and modeling research on the role of spinal reflexes during locomotion focuses on their contribution to the timing of swing and stance phases and to the production of muscle force in load bearing extensor muscles [Pang, M. Y., Yang, J. F., 2000. The initiation of the swing phase in human infant stepping: importance of hip position and leg loading. J Physiol 528 Pt 2, 389-404; Dietz, V., 2002. Proprioception and locomotor disorders. Nat Rev Neurosci 3 (10), 781-790; Ivashko, D. G., Prilutski, B. I., Markin, S. N., Chapin, J. K., Rybak, I. A., 2003. Modeling the spinal cord neural circuitry controlling cat hindlimb movement during locomotion. Neurocomputing 52-54, 621-629; Yakovenko, S., Gritsenko, V., Prochazka, A., 2004. Contribution of stretch reflexes to locomotor control: a modeling study. Biol Cybern 90 (2), 146-155; Ekeberg, O., Pearson, K., 2005. Computer simulation of stepping in the hind legs of the cat: an examination of mechanisms regulating the stance-to-swing transition. J Neurophysiol 94 (6), 4256-4268; Maufroy, C., Kimura, H., Takase, K., 2008. Towards a general neural controller for quadrupedal locomotion. Neural Netw 21 (4), 667-681; Donelan, J. M., Pearson, K. G., 2004. Contribution of sensory feedback to ongoing ankle extensor activity during the stance phase of walking. Can J Physiol Pharmacol 82 (8-9), 589-598; Frigon, A., Rossignol, S., 2006. Experiments and models of sensorimotor interactions during locomotion. Biol Cybern 95 (6), 607-627; Grey, M. J., Nielsen, J. B., Mazzaro, N., Sinkjaer, T., 2007. Positive force feedback in human walking. J Physiol 581 (1), 99-105]. The reflex contribution to load bearing has started to link positive force feedback to the underlying dynamics of the locomotor system [Prochazka, A., Gillard, D., Bennett, D., 1997. Positive force feedback control of muscles. J. of Neurophys. 77, 3226-3236; Geyer, H., Seyfarth, A., Blickhan, R., 2003. Positive force feedback in bouncing gaits? Proc. R. Soc. Lond. B 270, 2173-2183]. There appears to be no previous work that systematically expands on the idea of encoding principles of legged dynamics in the motor control system. While some of the muscle reflexes implemented in the human model were simple expedients to let it enter cyclic motions (trunk balance, swing-leg initiation), mainly the stance phase reflexes encoded principles of legged dynamics and control described previously, including compliant stance leg behavior [Blickhan, R., 1989. The spring-mass model for running and hopping. J. of Biomech. 22, 1217-1227; McMahon, T., Cheng, G., 1990. The mechanism of running: how does stiffness couple with speed? J. of Biomech. 23, 65-78; Geyer, H., Seyfarth, A., Blickhan, R., 2006. Compliant leg behaviour explains the basic dynamics of walking and running. Proc. R. Soc. Lond. B 273, 2861-2867], stabilization of segmented chains [Seyfarth, A., Günther, M., Blickhan, R., 2001. Stable operation of an elastic three-segmented leg. Biol. Cybern. 84, 365-382; Günther, M., Keppler, V., Seyfarth, A., Blickhan, R., 2004. Human leg design: optimal axial alignment under constraints. J. Math. Biol. 48, 623-646], and swing-leg retraction [Herr, H., McMahon, T., 2000. A trotting horse model. Int. J. Robotics Res. 19, 566-581; Herr, H., McMahon, T., 2001. A galloping horse model. Int. J. Robotics Res. 20, 26-37; Herr, H. M., Huang, G. T., McMahon, T. A., April 2002. A model of scale effects in mammalian quadrupedal running. J Exp Biol 205 (Pt 7), 959-967; Seyfarth, A., Geyer, H., 2002. Natural control of spring-like running—optimized self-stabilization. In: Proceedings of the 5th international conference on climbing and walking robots. Professional Engineering Publishing Limited, pp. 81-85; Seyfarth, A., Geyer, H., Herr, H. M., 2003. Swing-leg retraction: a simple control model for stable running. J Exp. Biol. 206, 2547-2555]. Based on these functional reflexes, the model not only converges to known joint angle and torque trajectories of human walking, but also predicts some individual muscle activation patterns observed in walking experiments. This match between predicted and observed muscle activations suggests that principles of legged mechanics could play a larger role in motor control than anticipated before, with muscle reflexes linking these principles into the neural networks responsible for locomotion.

In a preferred embodiment, the neuromechanical model of the invention has been implemented as a muscle reflex controller for a powered ankle-foot prosthesis. This embodiment is an adaptive muscle-reflex controller, based on simulation studies, that utilizes an ankle plantar flexor comprising a Hill-type muscle with a positive force feedback reflex. The model's parameters were fitted to match the human ankle's torque-angle profile as obtained from level-ground walking measurements of a weight and height-matched intact subject walking at 1 m/sec. Using this single parameter set, clinical trials were conducted with a transtibial amputee walking on level ground, ramp ascent, and ramp descent conditions. During these trials, an adaptation of prosthetic ankle work was observed in response to ground slope variation, in a manner comparable to intact subjects, without the difficulties of explicit terrain sensing. Specifically, the energy provided by the prosthesis was directly correlated to the ground slope angle. This study highlights the importance of neuromuscular controllers for enhancing the adaptiveness of powered prosthetic devices across varied terrain surfaces.

In order to produce a controller with the ability to adapt, the neuromuscular model with a positive force feedback reflex scheme as the basis of control of the invention was used as part of the control system for a powered ankle-foot prosthesis. The controller presented here employs a model of the ankle-foot complex for determining the physical torque to command at the ankle joint. In this model, the ankle joint is provided with two virtual actuators. For plantar flexion torque, the actuator is a Hill-type muscle with a positive force feedback reflex scheme. This scheme models the reflexive muscle response due to some combination of afferent signals from muscle spindles and Golgi tendon organs. For dorsiflexion torque, an impedance is provided by a virtual rotary spring-damper.

The parameters of this neuromuscular model were fitted by an optimization procedure to provide the best match between the measured ankle torque of an intact subject walking at a target speed of 1.0 m/sec, and the model's output torque when given as inputs the measured motion of the intact subject. The neuromuscular model-based prosthetic controller was used to provide torque commands to a powered ankle-foot prosthesis worn by an amputee. This control strategy was evaluated using two criteria. First, the controller was tested for the ability to produce prosthesis ankle torque and ankle angle profiles that qualitatively match those of a comparable, intact subject at a target level-ground walking speed. The second performance criterion was the controller's ability to exhibit a biologically-consistent trend of increasing gait cycle net-work for increasing walking slope without changing controller parameters. Detecting variations in ground slope is difficult using typical sensors, so a controller with an inherent ability to adapt to these changes is of particular value.

FIGS. 10A-C depict the physical system (FIG. 10A), a diagram of the drive train (FIG. 10B), and a mechanical model (FIG. 10C) for an exemplary embodiment of an ankle-foot prosthesis used in a preferred embodiment. The ankle-foot prosthesis used for this study is one in development by iWalk, LLC. This prosthesis is a successor to the series of prototypes developed in the Biomechatronics Group of the MIT Media Laboratory, which are described in U.S. patent application Ser. No. 12/157,727, filed Jun. 12, 2008, the entire disclosure of which has been incorporated by reference herein in its entirety. The prosthesis is a completely self-contained device having the weight (1.8 kg) and size of the intact biological ankle-foot complex. Seen in FIG. 10A are housing 1005 for the motor, transmission, and electronics, ankle joint 1010, foot 1015, unidirectional parallel leaf spring 1020, and series leaf spring 1025. Depicted in FIG. 10B are timing belt 1030, pin joint main housing 1035, motor 1040, ball screw 1045, ankle joint 1010, ball nut 1050 pin joint (series spring) 1055, and foot motion indicator 1060. Depicted in the mechanical model of FIG. 10C are parent link 1065, motor 1040, transmission 1070, series spring 1025, unidirectional parallel spring 1020, foot 1015, series spring movement arm r_(s) 1075, spring rest length 1080, and SEA 1085. The rotary elements in the physical system are shown as linear equivalents in the model schematic for clarity.

The ankle joint is a rolling bearing design joining a lower foot structure to an upper leg shank structure topped with a prosthetic pyramid fixture for attachment to the amputee's socket. The foot includes a passive low profile Flex-Foot™ (Osur™) to minimize ground contact shock to the amputee. A unidirectional leaf spring, the parallel spring, acts across the ankle joint, engaging when the ankle and foot are perpendicular to each other. It acts in parallel to a powered drive train, providing the passive function of an Achilles tendon. The powered drive train is a motorized link across the ankle joint as represented in FIG. 10B. From the upper leg shank end, it consists, in series, of a brushless motor, (Powermax EC-30, 200 Watt, 48V, Maxon) operating at 24V, a belt drive transmission with 40/15 reduction, and a 3 mm pitch linear ball screw. At this operating voltage, the theoretical maximum torque that can be generated by the motor through the drivetrain is approximately 340 Nm.

At the foot, the series spring, a Kevlar-composite leaf spring, connects the foot to the ball nut with a moment arm, r_(s), that is direction-dependent. Therefore, the effective rotary stiffness of the series spring, as evaluated by locking the drive train and exerting a torque about the ankle joint, is 533 N·m/rad for positive torque, and 1200 N·m/rad for negative torque, where positive torque (or plantar flexion torque) is that tending to compress the series spring as represented in FIG. 10C. The drive train and the series spring together comprise a series-elastic actuator (SEA) [G. A. Pratt and M. M. Williamson, “Series elastic actuators,” Proceedings on IEEE/RSJ International Conference on Intelligent Robots and Systems, Pittsburgh, pp. 399-406, 1995]. The arrangement of these components is shown schematically in FIG. 10C.

Sensors. A hall-effect angle sensor at the ankle joint is a primary control input, and has a range of −0.19 to 0.19 radians, where zero corresponds to the foot being perpendicular to the shank. Joint angle is estimated with a linear hall-effect sensor (Allegro A1395) mounted on the main housing. This sensor is proximate to a magnet that is rigidly connected to the foot structure so that the magnetic axis is tangent to the arc of the magnet's motion. As a result of this arrangement, the magnetic field strength at the sensor location varies as the magnet rotates past the sensor. Strain gauges are located inside the prosthetic pyramid attachment, allowing for an estimate of the torque at the ankle joint. Strain gauges located on the series spring permit sensing of the output torque of the motorized drive train, thereby allowing for closed-loop force control of the SEA. The motor itself contains Hall-effect commutation sensors and is fitted with an optical shaft encoder that enables the use of advanced brushless motor control techniques.

Microcontroller. Overall control and communications for the ankle-foot prosthesis are provided by a single-chip, 16-bit, DSP oriented microcontroller, the Microchip Technology Incorporated dsPIC33FJ128MC706. The microcontroller operates at 40 million instructions per second, with 128 kilo-bytes of flash program memory, and 16384 bytes of RAM. It provides adequate computation to support real time control.

Motor Controller. A second 16-bit dsPIC33FJ128MC706 was used as a dedicated motor controller. The high computation load and speed requirements of modern brushless motor control methodologies, along with task isolation from the main microcontroller's real time demands motivated this architecture. A high speed digital link between the main microcontroller and the motor microcontroller supplied virtually instantaneous command of the motor.

Wireless Interface. For development and data collection, a high speed serial port of the microcontroller is dedicated to external communications. This port may be used directly via cable or may have a wide variety of wireless communication devices attached. For the present study, the 500 Hz sensor and internal state information is telemetered over the serial port at 460 Kilobaud and transmitted via an IEEE 802.11g wireless local area network device (Lantronix Wiport).

Battery. All power for the prosthesis was provided by a 0.22 kg lithium polymer battery having a 165 Watt-Hour/kg energy density. The battery was able to provide a day's power requirements including 5000 steps of powered walking.

Optimal Mechanical Component Selection. Meeting the requirements for mass, size, torque, speed, energy efficiency, shock tolerance, and nearly silent operation is not a trivial task. Of particular importance is the modeling and optimization of the drive train for the production of the biological torques and motions of walking. Some effects of the motor selection, overall transmission ratio, series elastic spring, and parallel spring are described in S. K. Au, H. Herr, “On the Design of a Powered Ankle-Foot Prosthesis: The Importance of Series and Parallel Elasticity,” IEEE Robotics & Automation Magazine. pp. 52-59, September 2008.

Control Architecture. The purpose of the control architecture is to command an ankle torque appropriate to the amputee's gait cycle as determined from available sensor measurements of prosthetic ankle state. The controller determines the appropriate torque using a neuromuscular model of the human ankle-foot complex. In this model, a hinge joint, representing the human ankle joint, is actuated by two competing virtual actuators: a unidirectional plantar flexor which is a Hill-type muscle model, and a dorsiflexor which acts as either a bi-directional proportional-derivative position controller, or a unidirectional virtual rotary spring-damper, depending on the gait phase. A finite state machine maintains an estimate of the phase of the amputee's gait. Depending on this estimated gait phase, one or the other, or both of the virtual actuators produce torques at the virtual ankle joint. The net virtual torque is then used as the ankle torque command to the prosthesis hardware. Physical torque at the ankle joint is produced by both the motorized drive train and the parallel spring. The ankle angle sensor is used to determine the torque produced by the parallel spring, and the remaining desired torque is commanded through the motor controller.

Top Level State Machine Control. Top level control of the prosthesis is implemented by a finite state machine synchronized to the gait cycle. During walking, two states are recognized: swing phase and stance phase. Prosthesis sensor inputs (ankle torque as estimated from the pyramid strain gauges, ankle angle, and motor velocity) are continuously observed to determine state transitions. Conditions for these state transitions were experimentally determined. FIG. 11 depicts the operation of the state machine and the transition conditions. The dorsiflexor and plantar flexor virtual actuators develop torque depending on the gait state estimate from the state machine.

In FIG. 11, the swing state 1110 is visually depicted as SW 1120, and stance 1130 is divided into controlled plantar flexion (CP) 1140, controlled dorsiflexion (CD) 1150, and powered plantar flexion (PP) 1160. State transitions 1170, 1180 are determined using the prosthesis ankle torque, T_(P), as measured from the pyramid strain gauges, and prosthesis ankle angle, θ.

The transition to swing phase when the foot leaves the ground is detected by either a drop in total ankle torque to less than 5 N·m, as measured using the pyramid strain gauges, or a drop in measured ankle angle, θ, below −0.19 radians to prevent angle sensor saturation. Positive torque is defined as actuator torque tending to plantar flex the ankle, and positive angles correspond to dorsiflexion. To prevent premature state transitions, the ankle torque developed during the stance phase must exceed 20 N·m for these transitions to be enabled. In addition, a 200 ms buffer time provides a minimum time frame for the stance period. The transition to stance phase upon heel-strike is detected by a decrease in torque below −7 N·m as measured using the pyramid strain gauges.

A block diagram of an exemplary embodiment of a control system for an ankle-foot prosthesis according to this aspect of the invention is shown in FIG. 12. Depicted in FIG. 12 are neuromuscular model 1210 and biomimetic robotic limb 1270. Robotic limb 1270 includes torque control system 1200 and robotic limb joint 1260. Torque control system 1200 includes parallel spring model 1220, lead compensator 1230, friction compensator 1240, and motor controller 1250.

The prosthesis measured ankle state, (θ_(m), {dot over (θ)}_(m)) is used to produce a torque command from the neuromuscular model, τ_(d). This desired ankle torque is fed through a torque control system to obtain a current command to the prosthesis actuator. The three primary components of this torque control system are the feedforward gain K_(ff), lead compensator, and friction compensation term. The parallel spring contribution to prosthesis ankle torque, τ_(p), is subtracted from the desired ankle torque to obtain the desired actuator torque τ_(d,SEA). The closed-loop torque controller then enforces the desired actuator torque using the measured actuator torque, τ_(SEA). Finally, the friction compensation term produces an additional torque value, τ_(f), which is added to the output of the closed-loop torque controller.

FIGS. 13A-C are plots of prosthesis torque over one complete gait cycle (heel-strike to heel-strike of the same foot) for three walking conditions: level-ground (FIG. 13A), ramp ascent (FIG. 13B), and ramp descent (FIG. 13C). Shown for each are commanded torque mean 1305, 1310, 1315 (thin line)±standard deviation (dashed lines), and prosthesis torque, as estimated using the measured SEA torque contribution and angle-based estimate of the parallel spring torque contribution 1320, 1325, 1330 (thick line). Vertical (dash-dot) lines 1335, 1340, 1345 indicate the end of the stance phase.

Dorsiflexor Model. FIGS. 14A-C depict an exemplary embodiment of the musculoskeletal model as implemented on the prosthetic microcontroller, including the Hill-type muscle model and spring-damper attachments to the two-link ankle joint model (FIG. 14A), detailed Hill-type muscle model (FIG. 14B), and geometry of the muscle model skeletal attachment (FIG. 14C) including the variable moment-arm implementation and angle coordinate frame for the muscle model. Depicted in FIGS. 14A and 14C are mechanical representations of dorsiflexor (spring-damper) 1405, planar flexor (MTC) 1410, foot 1415, shank 1420, and heel 1425.

The dorsiflexor in FIG. 14A is the dorsiflexor actuator. It represents the Tibialis Anterior and other biological dorsiflexor muscles. This dorsiflexor is implemented as a virtual rotary spring-damper with a set point of [θ=0, {dot over (θ)}=0] and relation:

T _(dorsi) =K _(P) θ+K _(V){dot over (θ)}.   (1)

Here, K_(P) is the spring constant, and K_(V) is the damping constant, θ is the ankle angle and {dot over (θ)} is the ankle angular velocity. For the stance phase, the value of K_(P) was optimized along with other muscle model parameters to best match the stance phase behavior of the biological ankle for normal level-ground walking. The damping term, K_(V), was experimentally tuned for stance phase to 5 Nm-s/rad to prevent the forefoot from bouncing off the ground at foot-flat. Also during the stance phase, the dorsiflexor acts only to provide dorsiflexion torque, so to mimic the unidirectional property of biological muscles. Furthermore, when the torque generated by the dorsiflexor drops to zero during stance as a result of the foot becoming perpendicular to the shank, the dorsiflexor is disabled for the remainder of the stance phase. Therefore, the dorsiflexor only contributes to the torque production early in the stance phase, when human dorsiflexor muscles are known to play a significant role [J. Perry, Gait Analysis: Normal and Pathological Function, New Jersey: SLACK Inc., 1992, Chapter 4, pp. 55-57]. In the swing phase, the dorsiflexor acts as a position controller, driving the foot to the set-point [θ=0, {dot over (θ)}=0]. For this, a gain of K_(P)=220 N·m/rad and damping constant of K_(V)=7 N·m·s/rad provides for quick ground clearance of the foot early in the swing phase.

Plantar Flexor Model. The virtual plantar flexor in FIGS. 14A-C comprises a muscle-tendon complex, (MTC) which represents a combination of human plantar flexor muscles. The MTC is based on S. K. Au, J. Weber, and H. Herr, “Biomechanical design of a powered ankle-foot prosthesis,” Proc. IEEE Int. Conf. On Rehabilitation Robotics, Noordwijk, The Netherlands, pp. 298-303, June 2007, where it is discussed in further detail. It consists of a contractile element (CE) which models muscle fibers and a series element (SE) which models a tendon. The contractile element consists of three unidirectional components: a Hill-type muscle with a positive force feedback reflex scheme, a high-limit parallel elasticity, and a low-limit, or buffer, parallel elasticity. In series with the contractile element is the series element, which is a nonlinear, unidirectional spring representing the Achilles tendon. The attachment geometry of the muscle-tendon complex to the ankle joint model is nonlinear, complicating the calculation of torques resulting from the actuator force.

Plantar Flexor Series Elastic Element. The series elastic element (SE) operates as a tendon in series with the muscle contractile element as in [H. Geyer, A. Seyfarth, R. Blickhan, “Positive force feedback in bouncing gaits?,” Proc. R Society. Lond. B 270, pp. 2173-2183, 2003]. Taking ε as the tendon strain defined as:

$\begin{matrix} {{ɛ = \frac{l_{SE} - l_{slack}}{l_{slack}}},} & (2) \end{matrix}$

where l_(SE) is the length of the series element and l_(slack) is its rest length, the series element is specified to be a nonlinear spring described by H. Geyer, A. Seyfarth, R. Blickhan, “Positive force feedback in bouncing gaits?,” Proc. R Society. Lond. B 270, pp. 2173-2183, 2003:

$\begin{matrix} {F_{SE} = \left\{ {\begin{matrix} {{F_{\max}\left( \frac{ɛ}{ɛ_{ref}} \right)}^{2},} & {ɛ > 0} \\ {0,} & {ɛ \leq 0} \end{matrix},} \right.} & (3) \end{matrix}$

where F_(max) is the maximum isometric force that the muscle can exert. Following H. Geyer, A. Seyfarth, R. Blickhan, “Positive force feedback in bouncing gaits?,” Proc. R Society. Lond. B 270, pp. 2173-2183, 2003, this quadratic form was used as an approximation of the commonly-modeled piecewise exponential-linear tendon stiffness curve. This approximation was made so to reduce the number of model parameters.

Plantar Flexor Contractile Element. The contractile element (CE) of the plantar flexor virtual actuator, FIG. 14B, is a Hill-type muscle model with a positive force feedback reflex scheme. It includes active muscle fibers to generate force, and two parallel elastic components, as in H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication). The Hill-type muscle fibers exert a unidirectional force. This force is a function of the muscle fiber length, l_(CE), velocity, v_(CE), and muscle activation, A. The resulting force, F_(MF) is, as in H. Geyer, A. Seyfarth, R. Blickhan, “Positive force feedback in bouncing gaits?,” Proc. R Society. Lond. B 270, pp. 2173-2183, 2003, given by:

F _(MF)(l _(CE) , v _(CE) , A)=F _(max) f _(L)(l _(CE))f _(V)(v _(CE))A.   (4)

The force-length relationship, f_(L)(l_(CE)), of the Hill-type muscle is a bell-shaped curve given by:

$\begin{matrix} {{{f_{L}\left( l_{CE} \right)} = {\exp \left\lbrack {c{\frac{l_{CE} - l_{opt}}{l_{opt}w}}^{3}} \right\rbrack}},} & (5) \end{matrix}$

where, l_(opt) is the contractile element length, l_(CE), at which the muscle can provide the maximum isometric force, F_(max). The parameter w is the width of the bell-shaped curve, and the parameter c describes the curve's magnitude near the extremes of the bell, where:

f _(L)(l _(CE)=(1±w)l _(opt))=exp(c).   (6)

The force-velocity relationship, f_(V)(v_(CE)), of the CE is the Hill equation:

$\begin{matrix} {{f_{V}\left( v_{CE} \right)} = \left\{ {\begin{matrix} {\frac{v_{\max} - v_{CE}}{v_{\max} + {Kv}_{CE}},} & {v_{CE} < 0} \\ {{N + {\left( {N - 1} \right)\frac{v_{\max} + v_{CE}}{{7.56\mspace{14mu} {Kv}_{CE}} - v_{\max}}}},} & {v_{CE} \geq 0} \end{matrix},} \right.} & (7) \end{matrix}$

where v_(max)<0 is the maximum contractile velocity of the muscle, v_(CE) is the fiber contraction velocity, K is the curvature constant, and N defines the dimensionless muscle force (normalized by F_(max)) such that

N=f _(V)(v _(CE) =−v _(max)).   (8)

Following H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication), the force-length relationship for the high-limit parallel elasticity (HPE), set in parallel with the CE, is given by:

$\begin{matrix} {{F_{HPE}\left( l_{CE} \right)} = \left\{ {\begin{matrix} {{F_{\max}\left( \frac{l_{CE} - l_{opt}}{l_{opt}w} \right)}^{2},} & {l_{CE} > l_{opt}} \\ {0,} & {otherwise} \end{matrix},} \right.} & (9) \end{matrix}$

A low-limit, buffer parallel elasticity (LPE) is also included, based on H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication). This was given the form of the nonlinear spring:

$\begin{matrix} {{F_{LPE}\left( l_{CE} \right)} = \left\{ {\begin{matrix} {{F_{\max}\frac{\left\lbrack \frac{\left( {l_{CE} - {l_{opt}\left( {1 - w} \right)}} \right)}{l_{opt}} \right\rbrack^{2}}{\left( {w/2} \right)}},} & {l_{CE} \leq {l_{opt}\left( {1 - w} \right)}} \\ {0,} & {otherwise} \end{matrix}.} \right.} & (10) \end{matrix}$

Therefore, the total plantar flexor force is described by:

F _(CE) =F _(MF)(l _(CE) , v _(CE) , A)+F _(HPE) −F _(LPE).   (11)

Where F_(CE) is the force developed by the contractile element. Since the CE and SE are in series, the following equation holds: F_(CE)=F_(SE)=F_(MTC).

Reflex Scheme. The contractile element activation, A, is generated using the positive-force feedback reflex scheme shown in FIG. 15, as in [H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication); H. Geyer, A. Seyfarth, R. Blickhan, “Positive force feedback in bouncing gaits?,” Proc. R Society. Lond. B 270, pp. 2173-2183, 2003]. FIG. 15 depicts an exemplary embodiment of a reflex scheme for the virtual plantar flexor muscle, including the relationship among ankle angle, muscle force, and the plantar flexor component of ankle torque.

As depicted in FIG. 15, this feedback loop includes a stance phase switch for disabling the plantar flexor force development during the swing phase. During stance, the plantar flexor force, F_(MTC), is multiplied by a reflex gain Gain_(RF), delayed by Delay_(RF) and added to an offset stimulation, PRESTIM to obtain the neural stimulation signal. The stimulation is constrained to range from 0 to 1, and is low-pass filtered with time constant T to simulate the muscle excitation-contraction coupling. The resulting signal is used as activation in equation (4) with an initial value of PreA. In addition, a suppression gain, Gain_(SUPP), following H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication), was implemented to help prevent the two actuators from fighting each other during stance. Here, the torque generated by the dorsiflexor is reduced by either Gain_(SUPP)·F_(MTC) or until its value drops to zero.

Plantar Flexor Geometry and Implementation. Within the muscle model framework, the ankle angle, θ_(foot), is defined as shown in FIG. 14C. Using this angle as the input to the model, the length of the muscle-tendon complex is calculated as in H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication) by:

l _(MTC) =r _(foot)ρ(sin(φ_(ref)−φ_(max))−sin(θ_(foot)−φ_(max)))+l _(slack) +l _(opt).   (12)

where ρ is a scaling factor representing the pennation angle of the muscle fibers, and φ_(ref) is the ankle angle at which l_(CE)=l_(opt) under no load.

The fiber length, l_(CE) can be computed using l_(CE)=l_(MTC)−l_(SE), where l_(SE) is obtained from the inverse of (3) given the current value of F_(CE)=F_(SE)=F_(MTC) from the muscle dynamics. The fiber contraction velocity, v_(CE), can then be obtained via differentiation. This creates a first order differential equation governed by the dynamics of the neuromuscular model. This equation can be solved for F_(MTC) given the time history of θ_(foot) and initial condition. However, since integration is computationally more robust than differentiation, an integral form of this implementation was used to solve for F_(MTC), as described in H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication).

Given the attachment radius, r_(foot), and the angle, φ_(max), at which maximum muscle-tendon moment arm is realized, the relationship between F_(MTC) and the resulting plantar flexor contribution to ankle torque, T_(plantar), is given by

T _(plantar) =F _(MTC) cos(θ_(foot)−φ_(max))r _(foot) =F _(MTC) ·R(θ_(foot))   (13)

where R(θ_(foot)) is a variable moment arm resulting from the muscle attachment to the ankle joint model. This relationship is shown graphically in FIG. 15. Hence, the plantar flexor model can ultimately be treated as a dynamical system linking a single input, θ_(foot), to a single output, T_(plantar).

Neuromuscular Model Parameter Determination. The plantar flexor model is a lumped representation of all of the biological plantar flexor muscles. Likewise, the dorsiflexor represents all biological dorsiflexor muscles. In this work, joint and torque measurements were taken only at the ankle joint. As a result, the state of multi-articular muscles, such as the gastrocnemius, could not be accurately estimated. Therefore the plantar flexor was based upon the dominant monarticular plantar flexor in humans, the Soleus. Therefore, the majority of the plantar flexor parameters values are those reported in H. Geyer, H. Herr, “A muscle-reflex model that encodes principles of legged mechanics predicts human walking dynamics and muscle activities,” (Submitted for publication) for the Soleus muscle. Some parameters of the plantar flexor, as well as those for the dorsiflexor, however, were expected to either have been significantly affected by the lumped models, or were not well known from biology. These six parameters were fitted using a combination of a Genetic Algorithm and gradient descent to enable the neuromuscular model to best match the walking data of an intact subject.

Non-Optimized Parameter Values are shown in Table 6.

TABLE 6 l_(opt) [m] 0.04 w 0.56 l_(slack) [m] 0.26 c ln(0.05) v_(max) [l_(opt)/s] 6.0 N 1.5 ε_(ref) 0.04 K 5 PreA 0.01 ρ 0.5 T [s] 0.01 r_(foot) [m] 0.05 PreSTIM 0.01 Delay_(RF) [s] 0.02

Non-amputee Subject Data Collection. Kinetic and kinematic walking data were collected at the Gait Laboratory of Spaulding Rehabilitation Hospital, Harvard Medical School, in a study approved by the Spaulding committee on the Use of Humans as Experimental Subjects [H. Herr, M. Popovic, “Angular momentum in human walking,” The Journal of Experimental Biology, Vol. 211, pp 487-481, 2008]. A healthy adult male (81.9 kg) was asked to walk at slow walking speed across a 10 m walkway in the motion capture laboratory after informed consent was given.

The motion-capture was performed using a VICON 512 motion-capture system with eight infrared cameras. Reflective markers were placed at 33 locations on the subject's body in order to allow the infrared cameras to track said locations during the trials. The cameras were operated at 120 Hz and were able to track a given marker to within approximately 1 mm. The markers were placed at the following bony landmarks for tracking the lower body: bilateral anterior superior iliac spines, posterior superior iliac spines, lateral femoral condyles, lateral malleoli, forefeet and heels. Wands were placed over the tibia and femur, and markers were attached to the wands over the mid-shaft of the tibia and the mid-femur. Markers were also placed on the upper body at the following sites: sternum, clavicle, C7 and T10 vertebrae, head, and bilaterally on the shoulder, elbow, and wrist joints.

Ground reaction forces were measured using two staggered force plates (model no. 2222 or OR6-5-1, by Advanced Mechanical Technology Inc., Watertown, Mass., USA) which were incorporated into the walkway. The precision of these force plates measuring ground reaction force and center of pressure is approximately 0.1 N and 2 mm respectively. The force plate data was collected at 1080 Hz and synchronized with the VICON motion capture data. Joint torques were calculated from the ground reaction forces and joint kinematics using a modified version of a standard inverse dynamics model. Vicon Bodybuilder, by Oxford Metrics, UK was used to perform the inverse dynamics calculations.

Six trials were obtained for a slow level-ground walking speed (1.0 m/s mean) and a single trial was used to represent the target ankle and torque trajectories for this walking condition. The end of the stance phase was defined as the point in time when the joint torque first dropped to zero after the peak torque was reached in the gait cycle. This event occurred at 67% gait-cycle for the selected trial.

FIGS. 16A and 16B depict prosthesis-measured torque and angle trajectories during trials with an amputee subject compared to those of the biological ankle of a weight and height-matched subject with intact limbs. Shown in FIGS. 16A and 16B are ankle torque (FIG. 16A) and ankle angle (FIG. 16B) over a level-ground gait cycle from heel-strike (0% Cycle) to heel-strike of the same foot (100% Cycle). Plotted in FIGS. 16A and 16B are mean 1610, 1620 (thin line)±one standard deviation (dashed lines) for the prosthesis measured torque and angle profiles resulting from the neuromuscular-model control, and the ankle biomechanics 1630, 1640 (thick line) for a gait cycle of the weight and height-matched subject with intact limbs at the same walking speed (1 m/sec). Vertical lines indicate the average time of the beginning of swing phase 1650, 1660 (thin dash-dot line) for the prosthesis gait cycles and the beginning of the swing phase 1670, 1680 (thick dash-dot line) of the biological ankle.

Fitting of Model Parameters to Experimental Data via Optimization. The following parameters were chosen for tuning: F_(max), Gain_(FB), Gain_(SUPP), φ_(ref), and φ_(max). The goal of the parameter tuning was to find the parameter set that would enable the neuromuscular model to best match a biological ankle torque trajectory for a particular walking condition, given the corresponding biological ankle angle trajectory as input to the model. The cost function for the optimization was defined as the squared error between the biologic and model torque profiles during the stance phase, given the biological ankle angle trajectory, i.e.:

$\begin{matrix} {{Cost} = {\sum\limits_{t \in {STANCE}}\; {\left( {{T_{m}(t)} - {T_{bio}(t)}} \right)^{2}.}}} & (14) \end{matrix}$

where T_(m) is the torque output of the model, and T_(bio) is the biological ankle torque.

A Genetic Algorithm optimization was chosen to perform the initial search for optimal parameter values, and a direct search was included to pinpoint the optimal parameter set. The Genetic-Algorithm tool in Matlab was used to implement both optimization methods. The level-ground human walking data at the selected 1.0 m/s walking speed was used to provide the reference behavior for the optimization. The allowable range for each of the optimization parameters are shown in Table 7.

TABLE 7 Optimization Parameter Ranges Parameter (Units) Minimum Value Maximum Value F_(max) (N) 3000 7000 Gain_(FB) 0.6 1.5 K_(P) (N · m/rad) 20 250 Gain_(SUPP) 0 5 φ_(ref) (rad) 0.52 2.09 φ_(max) (rad) 1.40 2.44

The initial population was chosen by the optimizer. The parameter values obtained from the parameter optimization are shown in Table 8.

TABLE 8 Fitted Values of Neuromuscular Model Parameters F_(max) (N) 3377 Gain_(FB) 1.22 K_(P) (N · m/rad) 72.9 Gain_(SUPP) 0 φ_(ref) (rad) 1.49 φ_(max) (rad) 1.95

Results of the parameter optimization. As a verification of the optimization effectiveness, the optimization was run with the final parameters using the biological ankle angle profile as input to the neuromuscular model. A comparison of the resulting torque profile to the biologic torque profile is shown in FIG. 17.

As shown in FIG. 17, a comparison of the ankle moment profile from the intact biological ankle to that of the neuromuscular model with the biological ankle angle profile as the input and with optimized parameter values, are biological ankle moment (grey line) 1710, modeled dorsiflexor component (dash-dot line) 1720, modeled plantar flexor muscle component (thin line) 1730, and total neuromuscular model (plantar flexor and dorsiflexor) moment (dashed line) 1740. The neuromuscular model ankle moment matches the biological ankle moment almost exactly for most of the gait cycle.

Low-Level Torque Control. The physical torque actually produced at the ankle joint during stance phase is from the combined actions of the parallel spring and the motorized drive train. The rotary parallel spring stiffness is approximately linear in the range of operation, with a spring stiffness of 500 N·m/rad. Using this spring constant, the parallel spring contribution is predicted and subtracted from the desired ankle torque. The remaining torque must be produced by the motorized drive train.

The performance of the motorized drive train is improved by use of lead compensation, friction compensation and feed-forward techniques, as shown in FIG. 12. Experimental investigations of the open loop drive train dynamics were performed and used to implement these improvements [M. Eilenberg, “A Neuromuscular-Model Based Control Strategy for Powered Ankle-Foot Prostheses,” Master's Thesis, Massachusetts Institute of Technology, Cambridge, Mass., 2009]. The output torque versus commanded torque for level-ground walking, ramp ascent, and ramp descent is shown in FIGS. 13A-C. The prosthesis output torque was estimated using the strain gauge on the series spring for the SEA torque contribution, and the ankle angle-based parallel spring torque estimate for the parallel spring torque contribution.

Clinical Evaluation. The prosthesis was placed on the right leg of a healthy, active, 75 kg transtibial amputee. The subject was allowed time to walk on the prosthesis for natural adjustment. The wireless link to the prosthesis was used to record the walking data from these trials. During the level-ground walking trials, the subject was asked to walk across a 10 m long path. The target intended walking speed was set to 1.0 m/s to match that of the intact subject. The subject began walking approximately 5 m from the beginning of the pathway, and stopped walking approximately 3 m past the end of the path. Markers on the ground were used to note the beginning and end of the 10 m path. A stopwatch was used to verify the average walking speed for each trial by noting when the subject's center of mass passed over each of the markers. A total of 10 trials were captured. Trials with walking speeds within 5% of the target speeds were used for processing, resulting in 45 gait cycles. The subject was next asked to walk up an 11-degree, 2 m long incline at a self-selected speed. The subject started on level-ground approximately 2 m from the start of the incline and stopped approximately 1 m past the incline on a platform for 10 ramp-ascent trials. This same path was then navigated in reverse for 12 ramp-descent trials.

Data Analysis. The first three and last three gait cycles of the level-ground trials were assumed to be transients, and were therefore ignored. Each of the remaining gait cycles were re-sampled to span 1000 data points. Mean and standard-deviation trajectories were computed from the resulting data. For both ramp ascent and descent, the last step on the ramp was used as the representative gait cycle. Each selected gait cycle was re-sampled and averaged in the same manner as described for the level-ground trials.

The net work was calculated for each individual gait cycle by numerically integrating ankle torque over ankle angle from heel-strike to toe-off. Here the swing phase was ignored for the net work calculations. The average net work for each walking condition was then computed from the individual gait cycle net work values.

Results. Torque Tracking. A precondition of the present experiments was the ability of the ankle-foot prosthesis to actually produce the torques and speeds that would be commanded by the neuromuscular controller. This ability is demonstrated in FIGS. 13A-C, illustrating commanded torque versus measured output torque for level-ground walking, ramp ascent, and ramp descent.

Adaptation to Ground Slope. The evaluation of ground slope adaptation of the neuromuscular-model controlled prosthesis was confirmed by the clinical trial data of FIGS. 9 a-9 c. The numerically integrated data of those trials gave net work values (work loop areas) as follows:

Level-Ground 5.4 ± 0.5 Joules Ramp Ascent 12.5 ± 0.6  Joules Ramp Descent 0.1 ± 1.7 Joules

Comparison to a Biological Ankle. The purpose of this neuromuscular model is to represent the inherent dynamics of the human ankle-foot complex in a useful way. Therefore, one may evaluate the resulting prosthesis controller based upon its ability to mimic the human behavior. FIGS. 16A and 16B, discussed previously, show the level-ground walking torque and angle profiles from the prosthesis along with those of a weight and height-matched subject with intact limbs.

FIGS. 18A-C are plots of measured prosthesis torque-angle trajectories for three different walking conditions: level ground (FIG. 18A), ramp ascent (FIG. 18B), and ramp descent (FIG. 18C). Shown in FIGS. 18A-C, are mean 1810, 1820, 1830±one standard deviation. Arrows indicate forward propagation in time. The average prosthesis net work increases with increasing ground slope. This result is consistent with human ankle data from the literature [A. S. McIntosh, K. T. Beatty, L. N. Dwan, and D. R. Vickers, “Gait dynamics on an inclined walkway,” Journal of Biomechanics, Vol. 39, pp 2491-2502, 2006].

The measured ankle torque and ankle angle profiles of the prosthesis qualitatively match those of a comparable intact individual for level-ground walking. The differences observed are of a low order, and may reasonably be attributed to a number of factors, including atrophy and/or hypertrophy in the clinical subject's leg muscles resulting from amputation, differences in limb lengths, and perhaps the lack of a functional biarticular gastrocnemius muscle. In addition, the limited range of the prosthetic angle sensor prohibited the prosthesis from reaching the full range of motion of the intact ankle.

Ground Slope Adaptation. The neuromuscular control presented here exhibits an inherent adaptation to ground slope without explicit sensing of terrain. The increased ankle net work during ramp ascent, and the decreased ankle net work during ramp descent, as compared to that of level ground walking, is consistent with the behavior of an intact human ankle under the same conditions, according to data from [A. S. McIntosh, K. T. Beatty, L. N. Dwan, and D. R. Vickers, “Gait dynamics on an inclined walkway,” Journal of Biomechanics, Vol. 39, pp 2491-2502, 2006]. This variation of stance-phase positive net work across walking conditions indicates a slope-adaptive behavior that is emergent of the neuromuscular model. The ability of the neuromuscular model to produce these biomimetic changes in behavior suggests that the model embodies an important characteristic of the human plantar flexor muscles. In addition, it is anticipated that the model has the potential for speed adaptation. In an attempt to move faster, the wearer may push harder on the prosthesis. This additional force could cause the modeled reflex to command higher virtual muscle forces, resulting in greater energy output, and hence higher walking speeds.

While a preferred embodiment is disclosed, many other implementations will occur to one of ordinary skill in the art and are all within the scope of the invention. Each of the various embodiments described above may be combined with other described embodiments in order to provide multiple features. Furthermore, while the foregoing describes a number of separate embodiments of the apparatus and method of the present invention, what has been described herein is merely illustrative of the Appl. of the principles of the present invention. Other arrangements, methods, modifications, and substitutions by one of ordinary skill in the art are therefore also considered to be within the scope of the present invention, which is not to be limited except by the claims that follow. 

1. A method for controlling at least one robotic limb joint of a robotic limb, the method comprising the steps of: providing a neuromuscular model including a muscle model, muscle tendon lever arm and muscle tendon length equations and reflex control equations; receiving feedback data relating to a measured state of the robotic limb; determining, using the feedback data, and the muscle model, muscle tendon lever arm and muscle tendon length equations and reflex control equations of the neuromuscular model, at least one torque command to be applied to the robotic limb joint; and applying the at least one torque command determined by the neuromuscular model at the robotic limb joint.
 2. The method of claim 1, further including receiving the at least one torque command from the neuromuscular model with a torque control system, and wherein the at least one torque command is applied by the torque control system.
 3. The method of claim 2, wherein the robotic limb includes a sensor mounted to the robotic limb, and the feedback data is provided by the at least one sensor mounted at the robotic limb.
 4. The method of claim 3, wherein the neuromuscular model and the torque control system are configured to control the robotic limb, wherein the robotic limb is a leg, and wherein the method further includes, with a finite state machine synchronized to the leg gait cycle, receiving the feedback data from the at least one sensor and determining a gait phase of the robotic leg using the feedback data received.
 5. The method of claim 4, wherein the neuromuscular model and the torque control system are configured to control a robotic leg comprising an ankle joint.
 6. The method of claim 4, wherein the neuromuscular model and the torque control system are configured to control a robotic leg comprising a knee joint.
 7. The method of claim 5, wherein the robotic leg further comprises a knee joint.
 8. The method of claim 7, wherein the robotic leg further comprises a hip joint.
 9. The method of claim 3, wherein at least one sensor is an angular joint displacement and velocity sensor, a torque sensor, or an inertial measurement unit.
 10. The method of claim 3, wherein the feedback data includes joint angle and joint angular velocity measured by the at least one sensor.
 11. The method of claim 10, further including using the muscle tendon lever arm and muscle tendon length equations to determine a muscle moment arm and a muscle tendon length using the measured joint angle.
 12. The method of claim 11, further including, with the muscle model, determining muscle force using the muscle tendon length and a stimulation input.
 13. The method of claim 12, wherein the muscle model comprises a contractile element and a series-elastic element arranged in a muscle tendon unit.
 14. The method of claim 13, wherein the reflex control equations are configured in a local feedback loop, and further including, with the reflex control equations, receiving muscle force feedback from the muscle model and providing the stimulation input to the muscle model.
 15. The method of claim 14, wherein the muscle force feedback is positive force feedback.
 16. The method of claim 14, wherein the reflex control equations are configured to mimic the stretch reflex of an intact human muscle.
 17. The method of claim 2, wherein the torque control system includes a feed forward gain, a lead compensator and a friction compensator to adapt the at least one torque command and thereby obtain at least one current command.
 18. The method of claim 17, wherein the torque control system further includes a motor controller, and wherein applying the at least one torque command includes driving an actuator of the robotic limb joint with the at least one current command using the motor controller.
 19. The method of claim 18, wherein the torque control system further includes a parallel spring model. 